Quillen K-Theory: Lecture Notesmandal.faculty.ku.edu/M996QuillenKTheory/0TheKBooK.pdf · Quillen K-Theory: Lecture Notes Satya Mandal, U. of Kansas 2 April 2017 27 August 2017 6,

Quillen K-Theory: Lecture Notes

Satya Mandal, U. of Kansas

2 April 201727 August 2017

6, 10, 13, 20, 25 September 20172, 4, 9, 11, 16, 18, 23, 25, 30 October 20171, 6, 8, 13, 15, 20, 27, 29 November 2017

December 4, 6

2

Dedicated to my mother!

Contents

1 Background from Category Theory 7

1.1 Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Classical and Standard Examples . . . . . . . . . . . . . . . . . . . 9

1.1.2 Pullback, Pushforward, kernel and cokernel . . . . . . . . . . . . . . 12

1.1.3 Adjoint Functors and Equivalence of Categories . . . . . . . . . . . 16

1.2 Additive and Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.1 Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Exact Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4 Localization and Quotient Categories . . . . . . . . . . . . . . . . . . . . . 23

1.4.1 Quotient of Exact Categories . . . . . . . . . . . . . . . . . . . . . 27

1.5 Frequently Used Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5.1 Pullback and Pushforward Lemmas . . . . . . . . . . . . . . . . . . 29

1.5.2 The Snake Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.5.3 Limits and coLimits . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 Homotopy Theory 37

2.1 Elements of Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.1 Frequently used Topologies . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4 Construction of Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5 CW Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.5.1 Product of CW Complexes . . . . . . . . . . . . . . . . . . . . . . . 57

2.5.2 Frequently Used Results . . . . . . . . . . . . . . . . . . . . . . . . 59

3

4 CONTENTS

3 Simplicial and coSimplicial Sets 61

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Simplicial Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Geometric Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.1 The CW Structure on |K| . . . . . . . . . . . . . . . . . . . . . . . 68

3.4 The Homotopy Groups of Simplicial Sets . . . . . . . . . . . . . . . . . . . 73

3.4.1 The Combinatorial Definition . . . . . . . . . . . . . . . . . . . . . 73

3.4.2 The Definitions of Homotopy Groups . . . . . . . . . . . . . . . . . 77

3.5 Frequently Used Lemmms . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5.1 A Lemma on Bisimplicial Sets . . . . . . . . . . . . . . . . . . . . . 84

4 The Quillen-K-Theory 89

4.1 The Classifying spaces of a category . . . . . . . . . . . . . . . . . . . . . . 89

4.1.1 Properties of the Classifying Space . . . . . . . . . . . . . . . . . . 91

4.1.2 Directed and Filtering Limit . . . . . . . . . . . . . . . . . . . . . . 92

4.1.3 Theorem A: Sufficient condition for a functor to be homotopy equiv-alence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1.4 Theorem B: The Exact Homotopy Sequence . . . . . . . . . . . . . 101

4.2 The K-Groups of Exact Categories . . . . . . . . . . . . . . . . . . . . . . 110

4.2.1 Quillen’s Q-Construction . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2.2 The Higher K-groups of an Exact Category . . . . . . . . . . . . . 115

4.2.3 Higher K-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.3 Characteristic Exact Sequences and Filtrations . . . . . . . . . . . . . . . . 123

4.3.1 Additivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.3.2 The Prime Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.4 Reduction by Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.4.1 Extension closed and Resolving Subcategories . . . . . . . . . . . . 132

4.5 Dévissage and Localization in Abelian Categories . . . . . . . . . . . . . . 143

4.5.1 Semisimple Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.5.2 Quillen’s Localization Theorem . . . . . . . . . . . . . . . . . . . . 146

4.6 K-Theory Spaces and Reformulations . . . . . . . . . . . . . . . . . . . . . 149

4.7 Negative K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.7.1 Spectra and Negative Homotopy Groups . . . . . . . . . . . . . . . 150

CONTENTS 5

4.8 Suspension of an Exact Category . . . . . . . . . . . . . . . . . . . . . . . 151

4.8.1 Cofinality and Idempotent Completion . . . . . . . . . . . . . . . . 154

4.8.2 The K-theory Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 156

A Not Used 157

A.0.3 Some Jargon from [H] (SKIP) . . . . . . . . . . . . . . . . . . . . . 157

A.0.4 On SSet Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.0.5 Inclusion of Geometric Realizations . . . . . . . . . . . . . . . . . . 160

6 CONTENTS

Chapter 1

Background from Category Theory

Among the sources I used in this chapter are [HS, Sv, Wc1] and most importantly [HS].Importance of Category Theory, as a learning tool, is underrated. To me, Category Theoryis a way to learn how analogy works in mathematics, from one area to others. This is agood way to understand different areas of mathematics, just by learning one single area.First, I recall the basic definitions from [HS].

1.1 Preliminary Definitions

Definition 1.1.1. A Category C consists of the following:

1. A class of objects, denoted by Obj(C ).

2. For any two objected A,B in Obj(C ), a set of morphisms, from A to B, denoted byMor(A,B), more precisely byMorC (A,B). (Depending on the context, several othernotations are used, for Mor(A,B). One of them is Hom(A,B). Morphisms are alsocalled "arrows", "maps" and, other things appropriate to the specific context.)

3. To each triple A,B,C of objects, a law of composition

Mor(A,B)×Mor(B,C) −→Mor(A,C) sending (f, g) 7→ gf.

The above data satisfy the following Axioms:

A 1 Mor(A,B) are mutually disjoint.

A 2 For composeable morphisms f, g, h, the associativity holds: h(gf) = (hg)f .

7

8 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

A 3 For each object A, teethe is a morphism 1A : A −→ A such that, for composable f, g:1Af = f and g1A = g.

Given a category C , the following are some obvious definitions:

1. A morphism f : X −→ Y in C , is said to be an isomorphism, if there is a morphismg : Y −→ X such that fg = 1Y and gf = 1X . In this case, g would be called theinverse of f .

2. A morphism f : X −→ Y in C is called a monomorphism or injective, if for g1f =g2f =⇒ g1 = g2, for all pairs of morphisms g1, g2 : A −→ X. A morphism f : X −→Y in C is called a epimorphism or surjective, if for fg1 = fg2 =⇒ g1 = g2, for allpairs of morphisms g1, g2 : Y −→ B.

3. An object L in C would be called an initial object, if Mor(L,X) is a singleton, forall X in Obj(C . An object R in C would be called a final object (orterminal object),if Mor(X,F ) is a singleton, for all X in Obj(C . An object 0 in C would be calledzero object, if it is both initial and final object.

4. Injective objects and projective objects (will skip for now).

5. For a category C , define the opposite category C opp. Where Obj(E ) = Obj(C opp),and for two objects in C , MorC opp(A,B) = MorC (A,B).

Definition 1.1.2. A covariant functor F : C −→ D from a category C to another categoryD is defined by the following data:

1. To each object A ∈ C , it associates an object FA ∈ D .

2. For objects A,B, there is a set theoretic map

F : MorE (A,B) −→MorD(FA, FB)

3. Further, F (fg) = F (f)F (g) for any two composeable morphisms f, g.

A contravariant functor F : C −→ D is a covariant functor C opp −→ D .

Definition 1.1.3. Given two covariant functors F,G : C −→ D , a natural transformationτ : F −→ G, associates, to each object X in C , a morphism τX : F (X) −→ G(X), suchthat for any morphism f ∈Mor(X, Y ) the diagram

F (X)τX //

F (f)

G(X)

G(f)

F (Y ) τY// G(Y )

commutes.

1.1. PRELIMINARY DEFINITIONS 9

If τX is an isomorphism, for all objects X, the τ would be called a natural equivalence.

Likewise, define natural transformation and natural equivalence between two contravari-ant functors.

Example 1.1.4. Let C be a category and A ∈ Obj(C ). Then

X 7→MorC (A,X) is a covariant functor C −→ Sets.

andX 7→MorC (X,A) is a contravariant functor C −→ Sets.

They are called representable functors. These are, perhaps, the most significant examplesof functors.

Some special functors:

Definition 1.1.5. Let F : C −→ D be a functor.

1. F is said to be full, if

∀ X, Y ∈ Obj(C ) the map F : MorC (X, Y ) −→MorD(FX,FY ) is surjective.

2. F is called faithful, if

∀ X, Y ∈ Obj(C ) the map F : MorC (X, Y ) −→MorD(FX,FY ) is injective.

3. F is called fully faithful, if

∀ X, Y ∈ Obj(C ) the map F : MorC (X, Y ) −→MorD(FX,FY ) is bijective.

4. In particular, if F : Obj(C ) −→ Obj(D) is an "inclusion", then C is called a sub-category. So, the concept a full subcategory is to be distinguished from a non-fullsubcategory.

1.1.1 Classical and Standard Examples

We recall a some of the classical and Standard examples.

Examples 1.1.6. We recall a few examples and establish notations.

1. The category of sets will be denoted by Set. The objects of Set consists of the classof all sets. For two sets X, Y , a morphism f : X −→ Y is a set theoretic map.


2. The category of Topological spaces will be denoted by Top. The objects of Top con-sists of the class of all topological spaces. For two topological spacesX, Y ,Mor(X, Y )

is the set of all continuous maps f : X −→ Y . Note the empty set φ is the initialobject and singleton sets x are the terminal objects in Top.

In Topology, there are two more important categories:

(a) The category of Pointed Topological spaces will be denoted by Top•. Theobjects of Top• consists of the class of all pairs (X, x), where X is a topologicalspace and x ∈ X is a point, to be called the base point. For two objects(X, x), (Y, y), Mor((X, x), (Y, y)) is the set of all continuous maps f : X −→ Y ,with f(x) = y. Note, Top• has a zero object, namely (x, x) , where x isany singleton set.

(b) The category of Topological Pairs will be denoted by Top−Pairs. The objectsof Top−Pairs consists of the class of all pairs (X,U), where X is a topologicalspace and U ⊆ X is a topological subspace. For two objects (X,U), (Y, V ),Mor((X, x), (Y, y)) is the set of all continuous maps f : X −→ Y , with f(U) ⊆V .

We have commutative diagram of natural functors:

Top•i //

_

ι

Top

Top−Pairs

j

88

where the vertical functor ι is a full subcategory, while i, j may be called the "forgetfulfunctors". Note, i, j would not be full, while they are faithful.

3. The category of Groups will be denoted by Gr. The objects of Gr consists of the classof all groups. For two groups X, Y ,Mor(X, Y ) is the set of all group homomorphismsf : X −→ Y .

4. The category of Abelian Groups will be denoted by Ab. The objects of Ab consistsof the class of all abelian groups. For two abelian groups X, Y , Mor(X, Y ) is the setof all group homomorphisms f : X −→ Y .

5. The category of Rings, with unity and 0 6= 1 will be denoted by Ring. The objectsof Ring consists of the class of all Rings, with unity and 0 6= 1. For two Rings X, Y ,Mor(X, Y ) is the set of all ring homomorphisms f : X −→ Y , sending 1 7→ 1.


6. The category of commutative Rings, with unity and 0 6= 1 will be denoted by cRing.The objects of cRing consists of the class of all commutative Rings, with unityand 0 6= 1. For two commutative Rings X, Y , Mor(X, Y ) is the set of all ringhomomorphisms f : X −→ Y , sending 1 7→ 1.

7. The category of Schemes will be denoted by Sch. The objects of Sch consists of theclass of all Schemes. For two Schemes (X,OX), (Y,OY ), Mor ((X,OX), (Y,OY )) isthe set of all morphisms f : (X,OX) −→ (Y,OY ) of schemes (see [Hr, Sec II.2]).

8. Lat A be a commutative ring.

(a) Then, QCoh(A) or Mod(A) will denote the category of all A-modules. Ob-jects of Mod(A) is the class of all A-modules. Given two A-modules M,N , amorphism f : M −→ N is an A-linear homomorphism.

In particular, if A = k is a field, theMod(k) is the category of all vectors spaces.

(b) Coh(A) or fMod(A) will denote the category of all finitely generated A-modules.Objects of fMod(A) is the class of all finitely generated A-modules. Given twofinitely generated A-modules M,N , a morphism f : M −→ N is an A-linearhomomorphism.

In particular, if A = k is a field, the fMod(k) is the category of all vectorsspaces, of finite dimension.

(c) P(A) will denote the category of all finitely generated projective A-modules.Objects of P(A) is the class of all finitely generated Projective A-modules.Given two finitely generated A-modules P,Q, a morphism f : P −→ Q is anA-linear homomorphism.

9. Lat (X,OX) be a Scheme (see [Hr, Sec II.5]) .

(a) Then, QCoh(X) will denote the category of all quasi-coherent OX-modules.Objects of QCoh(X) is the class of all Quasi-Coherent OX-modules. Given twoQuasi-Coherent OX-modules F ,G, a morphism f : F −→ G is a morphism ofOX-modules.

(b) Coh(X) will denote the category of all coherentOX-modules. Objects of Coh(X)

is the class of all Coherent OX-modules. Given two Coherent OX-modules F ,G,a morphism f : F −→ G is a morphism of OX-modules.

(c) V ect(X) will denote the category of all locally free OX-modules. Objects ofV ect(X) is the class of all locally free OX-modules. Given two locally freeOX-modules F ,G, a morphism f : F −→ G is a morphism of OX-modules.


Above examples of categories, lead to a number of inclusion functors. Reader would bewell advised to think about them. However, we recall the following.

Example 1.1.7. Let A be a commutative ring.

1. Given a finitely generated projective A-module P , let P ∗ = HomA(P,A). Then,

P 7→ P ∗ defines a contravariant functor P(A) −→P(A)

Given a morphism ϕ : P −→ Q, and f ∈ Q∗, ϕ∗(f) is defined by the commutativediagram

Qf // A

Pϕ∗(f)

??

ϕ

OO

2. Composition of functors are functors. So, P −→ P ∗∗ is a covariant functor P(A) −→P(A).

3. For P ∈ Obj(P(A)) and for p ∈ P , define the evaluation map evp : P ∗ −→ A, byevp(f) = f(p). Then, evp ∈ P ∗∗. So, we have a homomorphism

ev : P −→ P ∗∗. In fact, it is and isomorphism.

Let IDP(A) denote the identity functor. Also, let D(P ) = P ∗∗ denote the doubledual functor. Then, ev : IDP(A) −→ D is a natural equivalence.

1.1.2 Pullback, Pushforward, kernel and cokernel

We define pullback and push forward.

Definition 1.1.8. Let Bg // X A

foo be two morphisms in a category C . Considerthe diagram:

Zϕ′

##

γ′

ζ

Y

ϕ //

γ

A

f

B g// X

We say that (Y, ϕ, γ), in short Y , is pullback of f, g , if

1. the rectangle commutes, i. e fϕ = gγ, and


2. Given morphisms ϕ′, γ′ such that fϕ′ = gγ′, there is a unique morphism ζ such thethe outer triangles commute.

We would write Pullback(f, g) := (Y, ϕ, γ), to mean that (Y, ϕ, γ) is the pullback of (f, g).

Likewise, the pushout of morphisms f : X −→ A, g : X −→ B in the category C isgiven by the commutative diagram:

Xf //

g

A

ϕ

ϕ′

B γ//

γ′ ++

Yζ

Z

That means, we say that (Y, ϕ, γ), in short Y , is the pushout of (f, g), if

1. the rectangle commutes, i. e ϕf = γg, and

2. Given morphisms ϕ′, γ′ such that ϕ′f = γ′g, there is a unique morphism ζ such thethe outer triangles commute.

We would write PushFwd(f, g) := (Y, ϕ, γ) or PushOut(f, g) := (Y, ϕ, γ), to mean that(Y, ϕ, γ) is the pushout of (f, g).

Remarks:

1. While the definition of pullback and pushout makes sense, the existence is not guar-anteed. But, if it exists, it is uniques up to isomorphism.

2. It is easy to see that pullback and pushout exists, in the following categories: (1) Gr,(2) Ab, (3) QCoh(A), where A is a commutative ring.

Very similar and retaliated is the concept of direct product and direct coproduct (sum).

Definition 1.1.9. Let C be a category and A,B ∈ Obj(C ). Consider the diagram:

Zϕ′

##

γ′

ζ

Y

ϕ //

γ

A

B


We say that (Y, ϕ, γ), in short Y , is the direct product of A,B , ifGiven morphisms ϕ′, γ′ such that fϕ′ = gγ′, there is a unique morphism ζ such the theouter triangles commute.

1. Again, while definition makes sense, existence of direct product is not guaranteed.But, it it exists, it is unique up to isomorphism. In that case, the direct product isdenoted by A×B.

2. If C has a terminal object 0T , then the direct product is, in deed, the pullback of thediagram

A

B // 0T

Likewise, the coproduct of morphisms A,B in the category C is given by the commutativediagram:

A

ϕ

ϕ′

B γ//

γ′ ++

Yζ

Z

That means, we say that (Y, ϕ, γ), in short Y , is the direct coproduct of (A,B), ifGiven morphisms ϕ′, γ′ such that ϕ′f = γ′g, there is a unique morphism ζ such the theouter triangles commute.

1. Again, while definition makes sense, existence of direct co-product is not guaranteed.But, it it exists, it is unique up to isomorphism. In that case, the direct co-productis denoted by A

∐B.

2. Depending on the context, direct co-product is also called the "direct sum". In thatcase, it would be denoted by A⊕B.

3. If C has an initial object 0I , then the direct product is, in deed, the pushout of thediagram

0I //

A

B

We define kernels.


Definition 1.1.10. Suppose C is a category with a zero object 0. Given X, Y ∈ Obj(C ),there is a unique "zero element" 0XY ∈Mor(X, Y ) given by the commutative diagram:

X //

0XY

0

Y

3 ∀ f ∈Mor(W,X) 0XY f = 0WY , and ∀ g ∈Mor(Y, Z) g0XY = 0XZ .

Suppose σ : X −→ Y is a morphism.

1. The Kernel of σ : X −→ Y is defined to be a morphism ι : K −→ X such that

(a) σι = 0 and

(b) Further,

∀ τ ∈Mor(L,X) with στ = 0 ∃ a unique τ0 ∈Mor(L,K) 3 τ = ιτ0.

(c) With my love for diagrams, the definitions is given by the diagram:

L τ//

0++

τ0 ∃!

X σ// Y

K0

33ι // Xσ // Y

2. The coKernel of σ is defined to be a morphism ζ : Y −→ C such that

(a) ζσ = 0 and

(b) Further,

∀ ψ ∈Mor(Y, Z) with ψσ = 0 ∃ a unique η ∈Mor(C,Z) 3 ηζ = ψ

(c) With my love for diagrams, the definitions is given by the diagram:

X0

++σ// Y

ζ// C

η∃!

Xσ //

0

33Yψ // Z


1.1.3 Adjoint Functors and Equivalence of Categories

We define Adjoint of a functors next.

Definition 1.1.11. Let F : C −→ D and G : D −→ C be two covariant functors. Wesay that F is a Left Adjoint of G and/or G is a Right Adjoint of F , if ∀ X ∈ Obj(C ), A ∈Obj(D), there is a bijection

η := ηXA : MorD(FX,A)∼−→MorC (X,GA), which is natural, in the sense that,

for any morphisms f : X −→ Y ∈ C and g : A −→ B ∈ D , the following two diagramscommute:

MorD(FX,A)

g∗

ηXA∼//MorC (X,GA)

G(g)∗

MorD(FX,B) ηXB

∼ //MorC (X,GB)

MorD(FX,A)ηXA∼//MorC (X,GA)

MorD(FY,A)

F (f)∗

OO

ηY A

∼ //MorC (Y,GA)

f∗

OO

In other words, the following two functors:

C opp ×D −→ Sets

(X,A) 7→MorD(FX,A)

(X,A) 7→MorC (X,GA)are naturally equivalent.

Likewise, we define Left/Right Adjoint of contravariant functors.

Definition 1.1.12. A functor F : C −→ D would be called an equivalence of categories,if there is a functor G : D −→ C such that FG is naturally equivalent to 1D and GF isnaturally equivalent to 1C (as in 1.1.3).

If follows, F is an equivalence of categories if and only if the following two conditionshold:

1. (Essentially Surjective): For any object Z ∈ Obj(D) there is an isomorphism FX∼−→

Z , for some object X ∈ Obj(C ).

2. (Fully Faithful): For all objects X, Y ∈ Obj(C ), the map

MorC (X, Y )∼−→MorD(FX,FY ) is a bijection.

We will recall the following example.

Example 1.1.13. Suppose A is a commutative ring. Then, there is an equivalence ofcategories

QCoh(A)∼−→ QCoh(Spec (A))

Proof. See [Hr, Sec II.5]. Note the functor above is a contra variant functor; not covariant.

1.2. ADDITIVE AND ABELIAN CATEGORIES 17

1.2 Additive and Abelian Categories

Definition 1.2.1. A category C is called an additive category, if (1) C has a zero object0, (2) For X, Y ∈ Obj(C ) the direct product X×Y exists, (3) For X, Y ∈ Obj(C ), the setMor(X, Y ) has the structure of an abelian group, such that the compositions is bilinear.That means, for objects X, Y, Z in C and f ∈Mor(X, Y ), g ∈Mor(Y, Z) the compositionmaps

g∗ : Mor(X, Y ) −→Mor(X,Z) h 7→ gh

f ∗ : Mor(Y, Z) −→Mor(X,Z) h 7→ hfare group homomorphims.

Definition 1.2.2. Suppose C ,D are two additive categories. A functor F : C −→ D iscalled a functor of additive categories, if for all objects X, Y of C , the map

F : MorC (X, Y ) −→MorD(FX,FY ) is a group homomorphism.

Proposition 1.2.3. Suppose C is an additive category. Then, for X, Y ∈ Obj(C ), thedirect sum exists and X × Y = X ⊕ Y .

Proof. See [HS, Prop II.9.1, pp. 76].

Proposition 1.2.4. Suppose C is an additive category and X, Y ∈ Obj(C ). Then, forf, g ∈Mor(X, Y ), then we have the following commutative diagram:

X ∆ //

f+g --

X ×X X ⊕X〈f,g〉Y

Proof. See [HS, Prop II.9.4, pp. 76]. We clarify, the vertical map (f, g) is, by definition,given by the diagram:

0 //

X

f

X //

g,,

X ⊕X

(f,g) ##Y


Lemma 1.2.5. Suppose C is an additive category and σ : X −→ Y is a morphism. Then,a morphism ι : K −→ X defines the kernel of σ if and only if ∀ A ∈ Obj(C ) the sequenceof groups

0 //Mor(A,K)ι∗ //Mor(A,X)

σ∗ //Mor(A, Y ) is exact.

Likewise, a morphism ζ : Y −→ C defines the coKernel of σ if and only if ∀ Z ∈ Obj(C )

the sequence of groups

0 //Mor(C,Z)ζ∗ //Mor(Y, Z) σ∗ //Mor(X,Z) is exact.

1.2.1 Abelian Categories

Definition 1.2.6. Suppose C is an additive category. Further, assume every morphismσ : X −→ Y has a kernel and a coKernel. Define the image and coImage

Im(σ) := ker (coKer(σ)) coIm(σ) := coKer (ker(σ))

Then, there is a natural map,

coIm(σ) −→ Im(σ).

Definition 1.2.7. (see [Sv]) Suppose A is an additive category. Then, A would be calledan Abelian Category, if

1. Every morphism σ : X −→ Y has a kernel and a coKernel.

2. For all morphisms σ : X −→ Y , the canonical morphism coIm(σ) −→ Im(σ) is anisomorphism.

Remark 1.2.8. For an arrow σ : X −→ Y in an abelian category A , coIm(σ)∼−→ Im(σ)

is an isomorphism. Because of this, throughout rest of these notes, we would usuallyidentify coIm(σ) and Im(σ), without further clarifications.

The definition in [HS, pp. 78], appears a little different:

Definition 1.2.9. (see [Sv]) Suppose A is an additive category. Then, A would be calledan Abelian Category, if

1. Every morphism σ : X −→ Y has a kernel and a coKernel.

1.2. ADDITIVE AND ABELIAN CATEGORIES 19

2. Every monomorphism is the kernel of its coKernel. Every epimorphism is the coK-ernel of its kernel.

3. Every morphism is expressible as composition of an epimorphism and a monomor-phism.

Proof of Equivalence of Two Definitions, follows from the following [HS, Prop 9.6,pp. 78]:

Proposition 1.2.10. Suppose A is an abelian category and σ : X −→ Y is a morphism.Then, we have the diagram:

Kµ // X

σ

η // I

νY

εC

where

µ = ker(σ)

ε = co ker(σ)

η = co ker(µ) =: coIm(σ)

ν = ker(ε) =: Im(σ)

Think I := image(σ)

σ = νη

Corollary 1.2.11. Suppose A is an abelian category.

1. A morphism σ : X −→ Y is isomorphism if and only if it is a monomorphism and anepimorphism.

2. Pullback and push forward exists.

Proof. See [HS, Cor 9.8] for the first one. Pullback exists, because it is a kernel:

Xϕ

##

_

P (ϕ, ψ)

99

%%

// X ⊕ Y // Z

Yψ

;;

?

OO

Example 1.2.12. Here are the main examples and comments:

1. Suppose A is a commutative ring. Then, Mod(A) = QCoh(A) and fMod(A) =

Coh(A) are Abelian categories. (Perhaps, the definition of the Abelian category wasmodeled after these two examples.)

2. More generally, let X be a scheme. Then, QCoh(X) and Coh(X) are Abelian cate-gories.


3. We remark, for a commutative ring A, the category P(A) is not necessarily anAbelian category. P(A) will be an Exact category, which we define next.

Lemma 1.2.13. Suppose A is an abelian category and f : M −→ N be an arrow. Then,f is an isomorphism if and only if f epi and mono.

Proof. Assume f is an isomorphism and g : N −→M be the inverse of f . So, fg = 1N andgf = 1M . Suppose fh = 0 for some arrow h : U −→ M . Then, h = 1Mh = gfh = g0 = 0.Hence, f is mono. likewise f is epi.

Now suppose f is both epi and mono. Since f is mono, ker(f) = 0 and likewisecoKer(f) = 0. From this it follows, coIm(f) = M , Im(f) = N and the induced mapis f : coIm(f) = M −→ Im(f) = N , which is an isomorphism by definition of Abeliancategories.

The proof is complete.

Lemma 1.2.14. Suppose A is an abelian category and f : X −→ Y is a morphism. Letν : I → Y be the image of σ and ν : X I be the corresponding epi morphism. Then,there is a factorization f = µν, where µ is epi and ν is mono. This factorization is unique,up to isomorphism, in the following sense:

I oν

o ι

Xf

//

µ?? ??

q

Y

J/

p

??

Given any other factorsation f = µ′ν ′, where µ′ is epi and ν ′ is mono, there is an isomor-phism ι : I

∼−→ J , such that the diagram commutes.

Proof. Suppose t : ker(f) → X be the kernel of f . Then, pqt = ft = 0. Since p is mono,qt = 0. Since µ : X −→ I is coKernel of t, there is an arrow ι : I −→ J , such that ιµ = q.Also, (pι)µ = pq = f = νµ. Since µ is mono, pι = ν. This establishes that the diagramcommutes.

Now, ι is epi, because so is q and ι is mono because so is ν. By Lemma 1.2.13, ι is anisomorphism.

1.3 Exact Category

We define Exact Categories as follows (see [Sm1]).

1.3. EXACT CATEGORY 21

Definition 1.3.1. Consider additive category E , together with a family of sequence ofmorphisms

X i // Y

p // // Z to be called conflations. (1.1)

We also say

0 // X i // Yp // Z // 0 is exact. (1.2)

The left part X // Y of such a conflation would be called an inflation and be denoted

as such. Likewise, the right part Y // // Z would be called a deflation and be deooted assuch.

Such an additive category E , together with such a family of conflations (1.1), would becalled an Exact Category, if the following conditions are satisfied:

1. In a conflation (1.1), the inflation i is the kernel of p and p is a coKernel of i.

2. Conflations are closed under isomorphisms.

3. inflations and deflations are closed under composition.

4. Any diagram Z Xoo i // Y of morphisms, with i inflation, can be completed toa push forward diagram

X

i // Y

Z

j//W

with j inflation.

5. Likewise, any diagram X // Z Ypoooo of morphisms, with p deflation, can be

completed to a pullback diagram

W

q

// Y

p

X // Z

with q deflation.

6. For all objects X, Y ∈ Obj(C ), the following

X // X ⊕ Y // Y is a conflation.

Further, an inflation is also called an admissible monomorphism and a deflation is alsocalled an admissible epimorphism.


Definition 1.3.2. Suppose E ,B are exact categories and F : B −→ E is a functor ofadditive categories. We say F is an exact functor, if it sends conflations to conflations.

Definition 1.3.3. Suppose E ,B are exact categories such that B ⊆ E be a full subcate-gory. We say that B is a fully exact subcategory of E , if B is closed under extensions inE , which means:

For a conflation (1.1) in E , if X,Z are isomorphic to an object in B, then so is Y .

1. This gives a structure of an exact category to B, by declaring a sequence in B exact,if it is isomorphic to an exact sequence in E .

2. Inclusion B ⊆ E preserves and detects conflations.

3. In this case, we also say that B ⊆ E is an extension closed exact subcategory.

Lemma 1.3.4. Every small exact category E can be embedded into an abelian category C ,as a fully exact, extension closed subcategory.

Proof. See [Q, pp 92]

Example 1.3.5. Here is a list of example (see [Sm1, Section 2.1.2]).

1. To start with, any abelian category is an exact category. Where a sequence

0 // Kι //M

p //// C // 0 (1.3)

is declared a conflations (or exact), if ker(p) = ι and co ker(ι) = p.

(a) So, for a ring R, the category R-mod, of all (left) R-modules is abelian (hencean exact) category, where conflations are defined as above (1.3).

(b) For a commutative ring A, QCoh(A) = Mod(A) and Coh(A) = fMod(A) areabelian categories, where conflations are defined as above (1.3).

(c) For a scheme X, the category OX-Mod (resp. QCoh(X)) of OX-modules (resp.quasi-coherent) OX-modules, is an abelian (hence an exact) category, whereconflations are defined as above (1.3).

(d) For a noetherian scheme X, the category Coh(X) of coherent OX-modules is anabelian (hence an exact) category, where conflations are defined as above (1.3).

1.4. LOCALIZATION AND QUOTIENT CATEGORIES 23

2. For a noetherian scheme X, the category V ect(X) of locally free sheaves form anexact category. If fact, V ect(X) is an exact subcategory of Coh(A).

In particular, for a commutative noetherian ring A, the category P(A) of finitelygenerated projective A-modules form an exact category.

3. Suppose E is an exact category. Let Ch(E ) be the category, whose objects are chaincomplexes:

(A•, d•) : · · · // Ai−1 // Ai di // Ai+1 // · · · 3 dd = 0.

Depending on the authors, subscripts are also use as follows:

(A•, d•) : · · · // Ai+1// Ai

di // Ai−1// · · · 3 dd = 0.

Morphisms in Ch(E ) are chain complex maps, given by commutative diagrams:

f • : (A•, d•) −→ (B•, ∂•) :

· · · // Ai−1

// Ai di //

f i

Ai+1 //

· · ·

· · · // Bi−1 // Bi

∂i// Bi+1 // · · ·

A sequence

(A•, d•) // (B•, d•) // (C•, d•)

is declared exact, if

∀ i ∈ Z Ai // Bi // Ci is exact.

Then, Ch(E ) is an exact category. Its full subcategory Chb(E ) of bounded chaincomplexes is also an exact subcategory. A complex (A•, d•) ∈ Chb(E ), if Ai =

0 ∀ i 0, i 0

1.4 Localization and Quotient Categories

The notes of Schlichting [Sm1] seems most helpful for this section. Inverting multiplicativesubsets S of a commutative rings A, is routinely covered in commutative algebra classes.However, this process extends to inversion of a class of morphisms in a category, as follows.


Definition 1.4.1. Suppose C is a category and S be a class of morphisms in C . (Onecan think of S as a subcategory, while we intend to invert all the morphisms in S.) Thelocalization of C with respect to S, is a functor ι : C −→ S−1C , where S−1C is a category,with the following universal property.

Given any functor F : C −→ D , such that F (f) is an isomorphism in D ,

∃ a unique functor ϕ : S−1C −→ D 3C ι //

F ""

S−1C

ϕ

D

commute.

As usual, existence on localization S−1C is not guaranteed, while if it exists, it is uniqueup to a unique equivalence.

We work toward giving a sufficient condition for existence.

Definition 1.4.2. Suppose C is a category and S be a class of morphisms in C . We saythat S satisfies a "calculus of right fractions" if the following three conditions hold:

1. The class S is closed under composition. The identity 1X is in S, for all X ∈ Obj(C ).

2. The first diagram below, can be completed to the second commutative diagram:

X

f

Y s// Z

with s ∈ S,W t //

g

X

f

Y s// Z

with t ∈ S (1.4)

3. Suppose f, g : X −→ Y be two morphisms in C and s : Y −→ Z, in S, such thatsf = sg. Then, there is a morphism t : W −→ X in S such that ft = gt.

If dual of these three conditions hold, then S is said to satisfy a "calculus of left fractions".If S satisfies both the conditions, then we say that S satisfies "calculus of fractions".

We remark that, barring set theoretic issues, S−1C exists, whenever S satisfies calculusof left or right fractions. We give the following version.

Lemma 1.4.3. Suppose C is a category and S is class of morphisms in C . Assumecodomains of morphisms in S has a set of isomorphism classes. Then, the localizationcategory S−1C exists.


Proof. We assume S satisfies calculus or right fractions. The objects of S−1C would bethe same as those of C . To define morphisms, for two objects X, Y , in view of Equation1.4, let

Pair(X, Y ) = (s, f) ∈MorC (U,X)×MorC (U, Y ) : Z ∈ Obj(C ), s ∈ S

Under the hypothesis, we can treat Pair(X, Y ) as a set, by restricting U to a set of represen-tatives of isomorphism classes of objects. Refer to the elements (s, f) ∈ Pair(X, Y ), by thediagrams X Usoo f // Y . Two such diagrams X Usoo f // Y , X V

too g // Yin Pair(X, Y ), are said to be equivalent, if

∃t : W −→ Us : W −→ V

either t or s ∈ S 3

Us

~~

f

X W

t

OO

s

Y

Vt

``

g

>> commute.

One checks that this is an equivalence relation on Pair(X, Y ). Let fs−1 denote the equiv-alence class of the diagram X Usoo f // Y and let

MorS−1C (X, Y ) := fs−1 : (s, f) ∈ Pair(X, Y ).

Let fs−1 ∈MorS−1C (X, Y ) and gt−1 ∈MorS−1C (Y, Z) be give by the diagrams X Usoo f // Y ,

Y Vtoo g // Z . To define compositions, by the conditions of calculus of right fractions,

we complete the following commutative diagram:

Wϕ //

τ

V

t

g // Z

X Usoo

f// Y

with τ ∈ S. Define (gt−1)(fs−1) := (gϕ)(sτ)−1.

Again, one checks that this composition is well defined. Further, it is clear that the functorι : C −→ S−1C satisfies the universal property. The proof is complete.

The quotient of additive categories by a subcategory is defined likewise, as follows.

Definition 1.4.4. Suppose A is an abelian category and B ⊆ A is an abelian subcategory.The quotient of A by B, is an abelian category A

B, together with an exact functor ι : A −→

AB

such that ι(X) = 0 for all X ∈ Obj(B), with the following universal property:

Given any exact functor F : A −→ D , where D is an abelian category, such thatF (X) = 0, for all objects in B,

∃ a unique additive functor ϕ :A

B−→ D 3

C ι //

F

AB

ϕ

D

commute.


If it the quotient AB

exists, then it is unique up to a unique equivalence However, existenceon localization is not guaranteed. Subsequently, we give a sufficient condition for existenceof the quotient.

Definition 1.4.5. Let A be an abelian category. A full subcategory B ⊆ A is said to bea Serre Subcategory, if for any conflation

K //M // // C in A K,C Obj(B) ⇐⇒M ∈ Obj(B) (up to isomorphisms).

That means, if and only if B is closed under subobjects, quotient objects and extensions.A Serre subcategory is an abelian subcategory.

Lemma 1.4.6. Let A be an abelian category (small) and B ⊆ A be a Serre subcategory.Then, the quotient category A

Bexists.

Proof. Suppose S be the set of morphisms f in A , such that ker(f), co ker(f) ∈ B. Then,S is satisfies the calculus of fractions:

1. Let f : X −→ Y , g : Y −→ Z be in S. We show that gf ∈ S. We have

ker(f)

ker(gf) // // ker(g) ∩ Im(f) is a conflation.

Since ker(g) ∈ Obj(B), ker(g) ∩ Im(f) ∈ Obj(B). Also, since ker(f) ∈ Obj(B),ker(gf) ∈ Obj(B).

By reversing the arrows, co ker(gf) ∈ Obj(B). So, S is closed under composition.

2. Now, we check the second condition of calculus of right fractions (Equation 1.4). LetY

f// Z Xsoo be arrows in B with s ∈ S. Consider the pull back diagram

X ×Z Yg //

t

X

s

Yf

// Z

By Lemma 1.5.1, ker(t) = ker(s) ∈ Obj(B). Now, extend the pullback diagram, toconflesions:

0 // X×ZYker(f)

g //

t

X

s

// co ker(g) //

s

0

0 // Yker(f) f

// Z // co ker(f) // 0

where t is induced by t, co ker(t) = co ker(t), and s is induced by s.


Now, from Snake Lemma 1.5.4,

ker(s) // ker(s) // co ker(t) // co ker(s) is exact.

Since ker(s), co ker(s) are in B, it follows that co ker(t) is in B. So, t ∈ S. So, thesecond condition, for right fractions, is satisfied by S.By reversing the arrows, it follows that the corresponding condition of left fractionsis satisfied.

3. To check that third condition, let f, g : M −→ N be two maps in A and s : N −→ Wbe a map is S, such that sf = sg. With F = f − g, we have sF = 0. It follows, thereis an arrow F0 : M −→ ker(s0 such that F = F0ι, as in the diagram

ker(s)

ι

M

0 ##

F0

;;

F// N

sW

This lead to

ker(F ) //M ×N ker(s) u //

ζ

ker(s)

ι

ker(F )

t//M

0''

F0

88

F// N

s

W

Claim t is in S. Clearly, ker(t) = 0 ∈ B. Now, co ker(t) = Im(F ) ⊆ ker(s) ∈ B.Hence, co ker(t) is in B. So, t ∈ S.Dual of this property is checked by reversing the arrows.

This establishes that S satisfies the calculus of fractions. Hence S−1A exists. Now, onechecks, that S−1A =: A

Bsatisfies the required universal property.

1.4.1 Quotient of Exact Categories

More generally, following [Sm2], we define quotient of exact categories.

Definition 1.4.7. Suppose E is an exact category and P ⊆ E be an extension closed fullsubcategory. An exact category Q, together with an exact functor ι : E −→ Q, would becalled a quotient of E by P, if ι satisfies the following conditions:

1. ι(P ) ' 0 for all P ∈ Obj(P).

2. The functor ι is universal, in the following sense:

Suppose D is an exact category ζ : E −→ D is an exact functor, such that ζ(P ) '0 ∀ P ∈ Obj(P). Then,

∃ Unique functor ϕ : Q −→ D

E ι //

ζ

Q

ϕ

D

commute.


Remarks:

1. The existence of such a quotient Q is not guaranteed. However, when it exists, it isunique up to isomorphism (functorial equivalence). In that case, we denote Q := E

P.

2. Readers may compare this definition with [Sm2, 1.14, 1.16].

3. It follows form the proof of Lemma 1.4.6, for a Serre subcategory of an abeliancategory A , the quotient A

Bexists.

4. Following Schlichting [Sm2], we give a sufficient condition for existence of such aquotient, as follows. This would be of some interest to us in Section 4.7.

Lemma 1.4.8. Suppose E is a small exact category and P ⊆ E be an extension closedfull subcategory. A morphism f : M −→ N in E would be called a weak isomorphism,if f = fnfn−1 · · · f1 is a finite composition of morphisms fi : Mi−1 −→ Mi is E , fori = 1, . . . , n, with M0 = M and Mn = N , such that

1. either fi is a deflation, with ker(fi) ∈ Obj(P),

2. or either fi is an inflation, with co ker(fi) ∈ Obj(P).

Let Σ denote the class of all weak isomorphisms. Further assume that Σ satisfies thecalculus of left or right fractions. Then, E

Pexists. Namely, the functor ι : E −→ Σ−1E

satisfy the universal property of the the quotient EP.

Proof. By Lemma 1.4.3, Σ−1E exists. A sequence in Σ−1E is declared a conflasion, if it isisomorphic to a conflation in E . This makes Σ−1E an exact category. Now let ζ : E −→ Dbe an exact functor, such that ζ(P ) ' 0 for all P ∈ Obj(P). If f ∈ Σ, then it follows thatζ(f) is an isomorphism. Hence, it follows that there is a unique functor ϕ : Σ−1E −→ Dsuch that ϕι = ζ. It is also clear that ϕ is exact.


Remark 1.4.9. Schlichting [Sm2] gives a sufficient condition, for the class Σ in (1.4.8)satisfies calculus of left or right fractions.

1.5 Frequently Used Lemmas

In this section, we list a frequently used Lemmas in K-Theory.

1.5. FREQUENTLY USED LEMMAS 29

1.5.1 Pullback and Pushforward Lemmas

Following two lemmas on pullback and push forward lemmas are used frequently in K-Theory.

The following two frequently used lemmas in K-Theorey.

Lemma 1.5.1. Suppose C is a category with zero. Let Xv // Z Y

hoo be two arrowin C . Let

Pp1 //

p2

X

v

Yh// Z

be the pullback.

1. Suppose (K, η) = ker(p2). Then, (K, p1η) = ker(v).

2. Let ι : K −→ X be the kernel of v. Then there is an induced arrow η : K −→ P ,given by the commutative diagram:

Kι

##

0

η

P

p1 //

p2

X

v

Yh// Z

Then, (K, η) = ker(p2). It is expressed better by the diagram:

K _η

K _ι

Pp1 //

p2

X

v

Yh// Z

One can write down the same, in terms of the kernels of the horizontal maps p1 andh.

Proof. See [HS, II Theorem 6.2].

Reversing the arrows, we have the following.

Lemma 1.5.2. Suppose C is a category with zero. Let X Zhoo v // Y be two arrow

in C . LetZ h //

v

X

i1

Yi2// A

be the pushout.


1. Suppose j : A −→ C be the coKernel of i2. Then, ji1 : X −→ C is the coKernel ofh.

2. Suppose ζ : X −→ D be the cockerel of h. Then, the following commutative diagramdefines a map β : A −→ C:

Z h //

v

X

i1 ζ

Yi2//

0 ++

A

β C

Then, (C, β) is the coKernel of i2. Same is better expressed by the diagram

Zh //

v

X

i1

ζ // C

Yi2// A

β// C

Proof. Reverse the arrow in the proof of (1.5.1).

1.5.2 The Snake Lemma

A key ingredients to K-Theory, is a classical result in Homological Algebra that is knownas the Snake Lemma. Before we state and prove Snake Lemma, we prove the followingelementary lemma.

Lemma 1.5.3. Suppose A is an abelian category. Consider the following diagram ofarrows:

Kf //M

g // C 3 fg = 0.

Let t : ker(g) → M be the kernel of g and s : Im(f) → M corresponds to the monomor-phism, corresponding to Im(f). Then, there is a unique arrow ι : Im(f) −→ ker(g) suchthat the diagram

Im(f) s //

ι

M

ker(g)

t//M

commute.

In fact, ι is a mono.


Proof. Since, t is mono, ι would also be mono, if it exists. Consider the diagram

K u// //

f,,Im(f)

s//M

g // C

Kv //

f

11ker(g) t //M

g // C

Since gsu = gf = 0 and u is epi, gs = 0. now, from definitions of kernel s, there is amorphism ι : Im(f) → ker(g), as required.

Most elementary version of Snake Lemma is stated for Modules over a ring. Thefollowing is one of the most general versions.

Lemma 1.5.4. Suppose A is an abelian category. Consider the commutative diagram

Ki //

f

Mj //

g

L //

h

0

0 // A ι// B

ζ// C

in A ,

where the rows are exact in A . Then, there is a natural exact sequence

ker(f) // ker(g) // ker(h) δ // coKer(f) // coKer(g) // coKer(h) ,

where the maps between the kernels, and between the coKernels are induced by the respectivemorphisms i, j, ι, ζ. The morphism δ is defined in the proof.

Proof. This proof is improvisation to some literature available in the net (due to JuliaGoedecke). We extend the above diagram and name the morphisms:

ker(f)ik //

_

fk

ker(g)jk //

_

gk

ker(h) _hk

K i //

f

Mj //

g

L //

h

0

0 // A ι//

fc

Bζ

//

gc

C

hccoKer(f) ιc

// coKer(g)ζc// coKer(h)

Construction of δ: Let (P, u, v) := Pullback(j, hk) and (Q,ϕ, η) := PushForward(ιc, fc),


which we display in the following diagram:

ker(j) i′ // P v // // _

u

ker(h) _hk

Ki0 // //

i0:: ::

i

22ker(j) i1 //M

j //

g

L //

h

##

0

0 // A ι//

fc

B

ϕ

ζ0// //

ζ,,coKer(ι)

// C

coKer(f)

η// Q

θ// // coKer(ι)

By properties of Exact categories (in this case Abelian Categories) or by Lemmas 1.5.1,1.5.2, η, u are injective and v, ϕ are surjective Also, by (1.5.1) and (1.5.2), we have ker(j) ∼=ker(v) =: i′ and coKer(η) = coKer(ι). We have

1. ϕgui′i0 = ϕgi = ϕιf = ηfcf = η0 = 0. Since i0 is surjective, ϕgui′ = 0. Sincei′ = ker(v), there is a map

α : ker(h) −→ Q 3P

v //

ϕgu

ker(h)

α||

Q

commute.

2. Likewise (dually), θ(ϕgu) = ζ0gu = hju = hhkv = 0v = 0. Since η = ker(θ), there isan arrow

β : P −→ coKer(f) 3

P

ϕgu

β

zzcoKer(f) η

// Q

commute.

3. Hence αv = ϕgu = ηβ. This implies θ(αv) = θ(ηβ) = 0. Since η = ker(θ), there isan arrow

γ : P −→ coKer(f) 3

Pγ

zzαv

coKer(f) η

// Q

4. Now, ηγi′ = αvi′ = 0. Since η is injective, γi′ = 0. Therefore,

∃ δ : ker(h) −→ coKer(f) 3

Pγ

αv

v // ker(h)δ

uucoKer(f) η

// Q

commute.


This completes the definition of the desired arrow δ. Now, we proceed to prove exactness.

1. Exactness at ker(g): By Lemma 1.5.3, there is an monomorphism Θ : Im(ik) −→ker(jk), such that sι = t, where s : ker(jk)) → ker(g) is the kernel of jk, andt : Im(ik) → ker(g) is the monomorphism, corresponding to the image of ik.To define the morphism in the opposite direction, consider the commutative diagramand label the morphisms:

ker(f) _fk

ik--

w// ker(jk)

τ

s// ker(g) _

gk

jk --ker(h) _

hk

Ki

22ϕ // // ker(j)

η //Mj

33 L

In this diagram the morphism τ is defined by the property of kernel ker(j), becausejgks = jkgks = 0. Now let (U, λ, µ) := PullBack(τ, ϕ). Consider the diagram:

Uλ // //

µ

ker(jk)

τ

s // ker(g) _gk

K

ϕ // //

i

22ker(j) η //M

Note λ is epi. We have

ϕµ = τλ =⇒ iµ = ηϕµ = ητλ = gksλ =⇒ giµ = 0 =⇒ ιfµ = 0 =⇒ fµ = 0,

because ι is mono. By property of kernel fk, we have

U µ //

ψ

Kf // A

ker(f) fk // K

f // A

So, gkikψ = ifkψ = iµ = gksλ =⇒ ikψ = sλ. Diagramatically,

Uµ // //

ψ

ker(jk) s // ker(g)

ker(f)ik

11u // // Im(ik)

t // ker(g)

By Lemma 1.2.14, µ = Im(sµ). Let ε = ker(sµ), as an arrow. Then, tuψε = 0 andhence uψε = 0. So, there is an arrow Ψ, as in the commutative diagram:

Uµ // //

ψ

ker(jk)

Ψ

s // ker(g)

ker(f)ik

11u // // Im(ik)

t // ker(g)

Inparticular tΨ = s


Since sΘ = t and s, t are mono, it follows ΦΘ = 1 and ΘΨ = 1. This completes theproof of exactness at ker(g).

2. Exactness at ker(h): We have the following commutative diagram:

ker(g)jk

&&

gk

w

""P v //

u

ker(h)

hk

Ki//

f

M

g

j// L

h

A ι// B

ζ// C

First, we want to prove δjk = 0. Since η is monic, we will prove ηδjk = 0, orηδvw = 0. However, ηδvw = ηγw = αvw = ϕguw = ϕggk = 0. Therefore byLemma 1.5.3, there is an arrow Ψ : Im(jk) → ker(δ) such that tΨ = s, wheret : ker(δ) → ker(h) is the kernel of δ and s : Im(jk) → ker(h) is the monomorphism,corresponding to Im(jk). Same is displayed in the commutative diagram:

Im(jk) s //

Ψ

ker(h)

ker(δ)

t// ker(h)

Now we proceed to define the inverse of Ψ. Consider the diagram

ker(f)ik //

_

fk

ker(g)jk //

_

gk

ker(h) _hk

λ // // coKer(jk) _

µ

K

i//M

j// L ν

// // F

coKer(gk)

p

33

o

Image(g)

where (F, µ, ν) := PushForw(λ, hk)

If follows, µ is mono and ν is epi. It follows νjgk = 0. So, νj factors through


coKer(gk). We further extend the above diagram:

ker(f)ik //

fk

ker(g)jk //

gk

ker(h)

hk

λ // // coKer(jk) _

µ

K

i//

f

M

g

j// L ν

// // F _

ξ

coKer(gk)

p

33

_

q

A ι

//

fc

Bψ // G

coKer(f)

ω

22

where (G, ξ, ψ) := pushFWD(p, q) = pushFWD(νj, g). Since q is mono, so is ξ.By the similar argument above, ψι factors through coKer(f). Now, we have thepushforward diagram, which defines an arrow τ .

A ι //

fc

B

ϕ

ψ

coKer(f)

ω,,

η// Q

τG

Now,

ωδv = τηδv = τηγ = ταv = τϕgu = ψgu = ξνju = ξνhkv = ξµλv

Since v is epi, ωδ = ξµλ. In summary,

ker(h) λ // //

δ

coKer(jk) _

µ

F _

ξ

coKer(f) ω// G

With t = ker(δ), we decompose:

ker(δ) t // ker(h) // //

δ..coIM(δ)

Ω

// coKer(f)

ω

ker(h)

λ// // coKer(jk)

µ// F

ξ// G


We have ξµλt = ωδt = 0. So, λt = 0. Since coIm(δ) = coKer(t), there is a uniquemorphism Ω, making the first rectangle commutative.

Again, since λt = 0, by property of kernel s, we have the commutative diagram, forsome arrow Θ:

ker(δ)

Θ

t // ker(h) e// //

δ..coIM(δ)

Ω

m// coKer(f)

ω

Im(jk)

s// ker(h)

λ// // coKer(jk)

µ// F

ξ// G

So, we have sΘ = t. Since tΨ = s, and s, t are mono, we have ΘΨ = Id and ΨΘ = Id.This establishes the exactness at ker(h).

3. Exactness at coKer(f): This follows from the dual argument of the proof of exact-ness at ker(h).

4. Exactness at coKer(g): This follows from the dual argument of the proof of exact-ness at ker(g).


1.5.3 Limits and coLimits

To be Inserted.

Chapter 2

Homotopy Theory

Much of the materials in this chapter would be available in standard textbooks on Topology.However, some of the not-so-standard materials were taken from [H].

2.1 Elements of Topological Spaces

We shall begin this chapter with the definition of topological spaces. The basic definitionsincluded in this section are omnipresent in the literature and the net. First and foremost,Topology is study of continuous functions. First, we define Topological spaces.

Definition 2.1.1. Suppose X is a nonempty set. A topology on X is a collection U ofsubsets of X, to be called open sets of X, such that

1. the empty set φ and X are open. That is φ,X ∈ U .

2. Union (finite or infinite) of open sets is open. That is if Ui : i ∈ I ⊆ U , then⋃i∈I Ui ∈ U .

3. Finite intersection of open sets is open. That is, if U1, U2, . . . , Un ∈ U , then ∩ni=1Ui ∈U .

When the set T is understood, we may just say X is a topological space.

Now, assume (X,U) is topological space. Here are few more definitions:

1. A subset C ⊆ X is called a closed set, if its complement Cc := X \ C is open.Therefore,

37

38 CHAPTER 2. HOMOTOPY THEORY

(a) Intersection (finite or infinite) of closed sets is closed.

(b) Finite union of closed sets is closed.

2. Let x ∈ X be a point. A subset U is called a neighborhood of x, if there is an opensubset V , such that x ∈ V ⊆ U . If U itself is open, we say U is an open neighborhoodof x.

3. A set of open sets B ⊆ U is called a basis of X, if given any open neighborhoodx ∈ U , of a point x, there is a an open set V ∈ B, such that x ∈ V ⊆ U .

So, given any open set U , we have U = ∪i∈IVi, where Vi ∈ B for all i ∈ I.

4. Given a subset A ⊆ X, then

A :=⋂C : A ⊆ C and C is closed

is called the closure of A. So, A is the smallest closed set containing A.

5. A subset D ⊆ X is called dense in X, if D ∩ U 6= φ, for all nonempty open sets U .It follows, D is a dense subset of X, if and only if the closure D = X.

Definition 2.1.2. Suppose (X,U) and (Y,V) and two topological spaces. A set theoreticmap f : X −→ Y is called a continuous function, if f−1(V ) ∈ U , for all V ∈ V .

1. If follows, the collection of all topological spaces, form a category (see Examples1.1.6). For topological spaces (X,U), (Y,V), Mor((X,U), (Y,V)) is defined to be theset of all continuous functions. This category was denoted by Top.

2. A continuous function f : (X,U) −→ (Y,V) is called homeomorphism, if f :

X −→ Y is bijective and f−1 : Y −→ X is continuous. So, homeomorphism f arethe "isomorphisms" in the category Top.

Examples 2.1.3. Here are some examples of topological spaces.

1. (Trivial Topology) Let X be a set and U = φ,X. Then, (X,U) is a topologicalspace. This is called the trivial topology on X and also the Indiscrete topology.

In this case, given any topological space (Y,V), any set theoretic function f : Y −→ X

is continuous.

2. (Discrete Topology) Let X be a set and P(X) be the power set of X, (i. e. P(X) isthe set of all subsets of X). Then, (X,P(X)) is topological space, to be called thediscrete topology on X. We have the following comments:

2.1. ELEMENTS OF TOPOLOGICAL SPACES 39

(a) In the discrete topology on X, all subsets of X are open (and also closed).

(b) B = x : x ∈ X be the set of all singleton subsets of X. Then, B is base forthis topology.

(c) Suppose (Y,V) is topological space and f : X −→ Y is any set theoretic function.Then, f : (X,P(X)) −→ (Y,V) is continuous.

3. (Induced Topology) Suppose (X,U) is a topological space and Y ⊆ X is a subset.Let

V = Y ∩ U : U ∈ U. Then, (Y,V) is a topological space.

4. (Usual Topology on Rn) For our purpose, the most important topology is theusual topology on Rn. For any x0 ∈ Rn and r > 0, define the open ball, of radius r,as

B(x0, r) = x ∈ Rn :‖ x− x0 ‖< r

A subset U ⊆ Rn is defined to be open, in usual topology, if U is union of open balls(possibly infinite union). Let U be the collection of all such open sets, Then (Rn,U)

is a topological space.

(a) Suppose X ⊆ Rn be a subset. Then, the usual topology on Rn induces atopology on X (as in (3)). We continue to call it the usual topology on X.The concept of continuity, taught in undergraduate classes, coincides with thecontinuity defined by Definition 2.1.2.

(b) Usual topology on Cn is defined in the same way.

(c) (Exercise) Prove that the set Q of rational numbers is a dense subset of R.

5. Examples from Algebra:

(a) (Affine Schemes) Suppose A is a commutative ring. Let Spec (A) be the set ofall prime ideals of A. For any ideal I ⊂ A, let V (I) = ℘ ∈ Spec (A) : I ⊆ ℘.The Zariski Topology on Spec (A) is defined to be the one, where closed sets areV (I), where I runs through all the ideals of A.

i. So, an open sets in Zariski Topology on Spec (A) are Spec (A)\V (I), whereI is an ideal.

ii. For f ∈ A, define D(f) = ℘ ∈ Spec (A) : f ∈ ℘. Then, B = D(f) : f /∈A forms a basis for this topology.


(b) (Projective Schemes) Suppose S = S0⊕S1⊕S2⊕· · · be a graded commutativering. Let S+ = S1 ⊕ S2 ⊕ · · · , to be called the irrelevant ideal. Let

Proj (S) = ℘ ∈ Spec (S) : ℘ is homogeneous prime ideal, S+ 6⊆ ℘

For any homogeneous ideal I ⊂ S, let V (I) = ℘ ∈ Proj (S) : I ⊆ ℘. TheZariski Topology on Proj (S) is defined to be the one, where closed sets areV (I), where I runs through all the homogeneous ideals of A.

i. So, an open sets in Zariski Topology on Proj (S) are Proj (S) \ V (I), whereI is a homogeneous ideal.

ii. For a homogeneous elemtn f ∈ Sn, define D(f) = ℘ ∈ Proj (S) : f ∈ ℘.Then, B = D(f) : f /∈ A forms a basis for this topology.

(We caution that, as "schemes" Spec (A) and Proj (S) would have more struc-tures, than what we gage above (see [Hr]))

The following are two fundamental concepts in topology.

Definition 2.1.4. Suppose X is a topological space. We say that is Hausdorff space,if any two distinct points x, y ∈ X, x 6= y, there are two open neighborhoods x ∈ Ux andy ∈ Uy such that Ux ∩ Uy = φ.

Clearly, Rn is a Hausdorff space.

Definition 2.1.5. Suppose X is a topological space. An open cover of X is a collectionof open sets Ui : i ∈ I, such that X =

⋃i∈I Ui.

1. X is said to be compact, if every open cover Ui : i ∈ I of X has a finite sub cover.This means, there are i1, . . . , in ∈, such that X =

⋃nj=1 Uij .

2. A subset Y ⊆ X is called the compact subset of X, if Y , with its induced topology,is a compact topological space.

We have two important exercise.

Exercise 2.1.6. Suppose X is a Hausdorff space and K ⊆ X is a compact subset. Then,K is closed. (This fails, if X is not Hausforff.)

Proof. We prove that Y := X \ K is open. Fix y ∈ Y . Then, ∀ x ∈ K, there are opensets Ux, Vx such that x ∈ Ux, y ∈ Vx and Ux ∩ Vx = φ. Since Ux ∩K : x ∈ K is an opencover of K, and K is compact K ⊆ Ux1 ∪ · · · ∪Uxn for finitely many x1, . . . xn ∈ K. WriteV = Vx1 ∩ · · · ∩ Vxn . Then, V is open, y ∈ V and K ∩ V = φ. So, y ∈ V ⊆ Y . Therefore,Y is open.

2.1. ELEMENTS OF TOPOLOGICAL SPACES 41

Exercise 2.1.7. Suppose K ⊆ Rn is a subset. Then, K is compact if and only if K isclosed and bounded.

Proof. Usually taught in the undergraduate Analysis courses and above.

2.1.1 Frequently used Topologies

We already defined Hausdorff Spaces and Compact spaces. Following is a list of a few morethat we may encounter in these notes.

Definition 2.1.8. Suppose X is a topological space.

1. The Topological space X is called is called locally compact, if X is Hausdorff and ifevery point x ∈ X has a compact neighborhood.

2. The Topological space X is called a Compactly generated space, if X is Hausdorffand if a subset A ⊆ X is closed ⇐⇒ A ∩ C is closed for every compact subset Cof X (see [W, pp. 18]). (The definition in [H, Def. 2.4.21] seems to differ a little.Perhaps, the goal in [H] was provide better functorial sense (see [H, Prop. 2.4.22].)

Following construction is useful, for our purpose.

Construction 2.1.9. Suppose X is Hausdorff space. We construct a new topology, de-noted by k(X), as follows: (1) Underlying set in k(X) is X, i.e. it is a topology on X.(2) A subset A ⊆ X is is closed in k(X) ⇐⇒ A ∩ C is closed for every compact subset Cof X.

We have the following observations:

1. k(X) → X is continuous. So, closed sets in X are closed in k(X).

2. k(X) is compactly generates.

3. Let K denote the category of compactly generated spaces. Then, there is an inclusionfunctor K → Top. Further,

X 7→ k(X) defines as functor Hausdorff −→ K .

where Hausdorff denotes the category of Hausdorff spaces. (Compare with [H, Prop.2.4.22].)


Given two topological spaces, X, Y , we would have to work with the function spacesMorTop(X, Y ). Following, is how MorTop(X, Y ) topologized.

Definition 2.1.10. Let X, Y be two topological spaces. Then, MorTop(X, Y ) may becalled a function space. We define two topologies on MorTop(X, Y ).

1. First, we define the Compact-Open Topology.

For any two subsets A,B ⊆ X, let

C(A,B) = f ∈MorTop(X, Y ) : f(A) ⊆ B.

B = C(K,U) : K is compact subset and U is open subset of X.

The Compact-Open Topology on MorTop(X, Y ) is the topology generated by B.Unless stated otherwise, MorTop(X, Y ) together with Compact-Open Topology willbe devoted by MorTop(X, Y ).

(a) That means, Compact-Open Topology on MorTop(X, Y ) is the smallest topol-ogy on MorTop(X, Y ), containing B. One can see an open set in this topologyis union of finite intersection of members of B.

(b) Let x0 ∈ X. The the evaluation map evx0 : MorTop(X, Y ) −→ Y is continuous.Because, for an open set U of Y , ev−1

x0(U) = C(x0, U).

(c) The following Lemma is about Hausdorff property:

Lemma 2.1.11. If X, Y is Hausdorff, then the Compact-Open Topology onMorTop(X, Y ) is also Hausdorff.

Proof. Let f, g ∈MorTop(X, Y ) and f 6= y. So, there is x0 ∈ X such that y1 :=

f(x0) 6= g(x0) =: y2. Since Y is Hausdorff, there are open neighborhood y1 ∈ V1

and y2 ∈ V2 such that V1∩V2 = φ. It follows C(x0, V1)∩C(x0, V2) = φ andf ∈ C(x0, V1), g ∈ C(x0, V2). The proof is complete.

2. For topological spaces X, Y , F(X, Y ) would denote the compactly generated topo-logical space k (MorTop(X, Y )).

2.2 Homotopy

The set of all real numbers would be denoted by R. Denote, the unit sphere and the unitdisc/Ball, as follows

Sn =x ∈ Rn+1 :‖ x ‖= 1

, Dn = x ∈ Rn :‖ x ‖≤ 1 .

2.2. HOMOTOPY 43

For n = 0, we let D0 = 0 and S−1 = φ. Also, write I := [0, 1], for the unit interval. So,we have inclusions Sn−1 ⊆ Dn ⊆ Rn. Note Dn ∼= In are homeomorphic.

Definition 2.2.1. We give variety of definitions and notations, related to homotopy.

1. Let X, Y be topological spaces and f, g : X −→ Y be two continuous maps. We sayf is homotopic to g, if

∃ a continuous map H : X × I −→ Y 3 ∀ x ∈ X

H(x, 0) = f(x)

H(x, 1) = g(x).(2.1)

Homotopy is an equivalence relation. We define the Homotopy Category H omoTop

of topological spaces, as follows:

(a) The objects of H omoTop are the topological spaces.

(b) For topological spaces X, Y , and continuous maps f : X −→ Y , let f denotethe homotopy equivalence class of f . The set of moronism in H omoTop isgiven by

MorH omoTop(X, Y ) := Mor(X, Y ) := f : f : X −→ Y is a continuous map

2. Now we consider Homotopy of maps in the category Top•. Assume X, Y are pointedspaces, with base points x0 ∈ X and y0 ∈ Y . Assume f, g : X −→ Y preservebase points. (In other words, f, g : (X, x0) −→ (Y, y0) are two maps in the categoryTop•.) Then, we say that f is homotopic to g, if

∃ continuous map H : X × I −→ Y 3

H(x, 0) = f(x) x ∈ XH(x, 1) = g(x) x ∈ XH(x0, t) = y0 t ∈ I

(2.2)

We say that H is a base point preserving homotopy. This homotopy of base pointpreserving maps is also an equivalence relation. Define the Homotopy CategoryH omoTop• of pointed topological spaces, in the same way as H omoTop.

3. Now we consider Homotopy in th category TopPairs, of topological pairs (X,X0).Recall, a map f : (X,X0) −→ (Y, Y0) in TopPairs is a continuous map f : X −→ Y

such that f(X0) ⊆ Y0.

Given two maps f, g : (X,X0) −→ (Y, Y0) of topological pairs, f is defined to homo-topic to g, if

∃ continuous map H : X × I −→ Y 3

H(x, 0) = f(x) x ∈ XH(x, 1) = g(x) x ∈ XH(x, t) ∈ Y0 x ∈ X0, t ∈ I

(2.3)


In this case also, homotopy of topological pairs is an equivalence relation. Definethe Homotopy Category H omoTopPair of topological pairs, in the same way asH omoTop. As for the original categories, we have a commutative diagram offunctors:

H omoTop•i //

ι

H omoTop

H omoTopPairj

66

Here the vertical functor represents a full subcategory.

4. Fix a base point ∗ ∈ Sn, say ∗ = (1, 0, . . . , 0) ∈ Sn . Given a pointed space (X, x0),πn(X, x0) would denote the set of all equivalences classes of maps (Sn, ∗) −→ (X, x0).That means,

πn(X, x0) := MorH omoTop• ((Sn, ∗), (X, x0)) is the representable functor,

from H omoTop• −→ Set corresponding to the object (Sn, ∗).(2.4)

This set πn(X, x0) is referred to as the nth-homotopy set (at x0). This study of thesesets πn(X, x0) constitutes the Classical Homotopy Theory.

Remark. The interpretation (2.4), of πn, as a representable functor, provides a clue,that one can replace (Sn, ∗), by any sequence (Tn, ∗) of pointed spaces, and developa theory considering the sets τn(X, x0) := MorH omoTop• ((Tn, ∗), (X, x0)), parallel tothe Classical Homotopy Theory.

(a) So, π0(X, x0) is the set of all path connected components of X.

(b) Note, π1(X, x0) is called the Fundamental Group of X.

(c) For fixed n, the association (X, x0) 7→ πn(X, x0) can also be considered ascovariant functor from Top• −→ Set, and we have a commutative diagram offuntors:

Top•πn

&&H omoTop• πn

// Set

More explicitly, given continuous maps

ϕ : (X, x0) −→ (Y, y0) denote πn(ϕ) : πn(X, x0) −→ πn(Y, y0)

2.2. HOMOTOPY 45

(d) Subsequently, we explain that, for n ≥ 1, there are group structures on πn(X, x0).In fact, the group structure on πn(X, x0) would be abelian groups, for n ≥ 2.

5. Let ∂In ⊆ In denote the boundary of In. Consider the maps (In, ∂In) −→ (X, x0)

of pairs of topological spaces and homotopy.

(a) If follows:

Lemma 2.2.2. The set πn(X, x0) is in bijection with the set of equivalenceclasses of maps (In, ∂In) −→ (X, x0) of pairs of topological spaces. In otherwords, there is a bijection

πn(X, x0)∼−→MorHomoTopPair ((In, ∂In), (X, x0)) .

Outline of the Proof. First, consider the quotient map λ : In In

∂In. Let

∗ = [∂In] ∈ In

∂Inbe the base point. Then,

i. There is a homeomorphism θ : (Sn, ∗) ∼−→(In

∂In, ∗).

ii. There is a bijectionMorTop•

((In

∂In, ∗), (X, x0)

) ∼−→MorTopPair ((In, ∂In), (X, x0))

sending f 7→ fλ

iii. A continuous map f : (Sn, ∗) −→ (X, x0) in Top•, gives rise to a mapfθ−1λ : (In, ∂In) −→ (X, x0).

This defines a bijection

HomTop• ((Sn, ∗), (X, x0)) −→ HomTopPair ((In, ∂In), (X, x0))

So, we established something stronger. one further check that this bijectionrespects homotopy.

(b) Using this description, it follows that, for n ≥ 1, the set πn(X, x0) has a groupstructure, which is abelian if n ≥ 2.For f, g : (In, ∂In) −→ (X, x0) the product is given by the class of

(fg) : (In, ∂In) −→ (X, x0) 3 (fg)(t1, . . . , tn) =

f(2t1, t2, . . . , tn) 0 ≤ t1 ≤ 1

2

g(2t1 − 1, t2, . . . , tn) 12≤ t1 ≤ 1

One checks, that this induces a well defined product on πn(X, x0), for n ≥ 1.

(c) Obviously, the constant map is the identity, which needs a proof, as well.


(d) For n ≥ 2, the groups πn(X, x0) is abelian. This also needs proof, that used thesecond coordinate.

6. Suppose ϕ, ψ : (X, x0) −→ (Y, y0) are homotopic, in the sense of (2.2), then

πn(ϕ) = πn(ψ) : πn(X, x0) −→ πn(Y, y0) ∀ n ≥ 0.

Proof. This is reinterpretation of the fact (2.4), that

πn(X, x0) := MorH omoTop• ((Sn, ∗), (X, x0)) is the representable functor.

7. Suppose X is a path connected space and x0, x1 ∈ X. Then, πn(X, x0)∼−→ πn(X, x1).

To establish this, in two steps:

(a) As always, I = [01]. Let γ : I −→ X be a path, with γ(0) = x0 and γ(1) = x1.Also, let f : (In, ∂In) −→ (X, x0) is a map in TopPair. Define

µ : In −→ [0, 1] by µ(t) = max| ti − .5 |, where t = (t1, . . . , tn) ∈ In.

Note µ(t) ≤ 12∀ i ∈ In. Now define

f ∗ γ : (In, ∂In) −→ (X, x1) by f ∗ γ(t) =

f(2t− .5) if µ(t) ≤ 1

4

γ(4µ(t)− 1) if 14≤ µ(t) ≤ 1

2

Note, f ∗ γ : (In, ∂In) −→ (X, x1) is a map in TopPair. One checks, if f, g arehomotopic in TopPair, then f ∗ γ is homotopic to .f ∗ γ. This establishes thatγ induces a map

Ψγ : π0(X, x0) −→ π(X, x1).

In fact, Ψγ is a homomorphism, for n ≥ 1.

(b) In fact, Ψγ is a bijection (or isomorphism for n ≥ 1).

Let γ′(t) = γ(1− t). Then γ′ is a path from x1 to x0. One checks that Ψγ andΨγ′ are inverse of other.

8. Now, suppose f, g : X −→ Y are homotopic, in Top only, by a homotopy H :

X × I −→ Y . Then,

For, x0 ∈ X α(t) = H(x0, t) : I −→ Y, is the trajectory of x0 y0 := f(x0) 7→ y1 = g(x0).

Then, conjugation by α defines an isomorphism

hα : πn(Y, f(x0)) −→ πn(Y, g(x0))

2.3. FIBRATIONS 47

It follows, that the diagram

πn(X, x0)πn(f) //

πn(g) &&

πn(Y, y0)

hαo

πn(Y, y1)

commute.

2.3 Fibrations

I recovered three definitions of Fibrations [H, S, W]. However, they are about homotopylifting property. We first recall the one from [H].

Definition 2.3.1 (See [H]). Let J denote the set of inclusions Dn → Dn × I sendingx 7→ (x, 0). A map f : X −→ Y is called a fibration, if f has right lifting property, withrespect to J, in the sense of the diagram,

Dn _

// X

f

Dn × I //

ϕ;;

Y

Equivalently,

In _

// X

f

In × I //

ϕ;;

Y

meaning, given the outer commutative diagram of continuous maps, there is a continuousmap ϕ, making all the triangles commute. Such fibrations are often known as weakfibrations (or Serre fibration). (While we give two more definitions from [S, W], bydefault "fibration", would mean a weak fibration.)

Corollary 2.3.2. The constant map X −→ x0 is trivially a fibration. This fact is statedas, every topological space X is fibrant.

Proof. Suppose F0 : D −→ X. Define ϕ : D × I −→ X, by ϕ(x, t) = F0(x). So, thediagram

Dn _

// X

f

Dn × I //

ϕ::

x0commute.

We recall two more definitions, from Spanier [S] and Whitehead [W].

Definition 2.3.3. Let p : E −→ B be a continuous map.

1. (See Spanier [S, pp. 66]): We say p is a topological fibration, if p has homotopylifting property, with respective to every space X:

X _

i0

// E

p

X × I

;;

F// B

, which is called homotopy lifting property.


A topological fibration is also called a Hurewicz fibration.

2. (See [W, pp. 29]): Under the same notations, as above, p is called a fibration,which we may call White fibration, the homotopy lifting property holds for X in thecategory K of Compactly generated topological space (see [H, pp. 18]).

3. Therefore,

f is topological fibration =⇒ f is White fibration =⇒ f is Weak fibration.

The following [H, Lemma 2.4.16] is of our primary interest.

Lemma 2.3.4. Suppose p : X −→ Y is a weak fibration, and x0 ∈ X. Let F = p−1(p(x0))

and y0 = p(x0). Let ι : F → X denote the inclusion map. Then, there is a long exactsequence

· · · // πn+1(F, x0)ι∗ // πn+1(X, x0)

p∗ // πn+1(Y, y0)

∂n+1 // πn(F, x0)ι∗ // πn(X, x0)

p∗ // πn(Y, y0)

// · · · · · · · · ·

// π1(F, x0) // π1(X, x0) // π1(Y, y0)

// π0(F, x0) // π0(X, x0) // π0(Y, y0)

(2.5)

Here ∂n is group homomorphism, ∀ n ≥ 2. For n = 0,

1. π0(X, x0) is to be understood to be a pointed set, where the path component of x0

is the base point.

2. By kernel, we mean the inverse image of the base point.

3. By exactness, we mean that the image is same as the kernel.

Proof. See [S, Theorem 7.2.10] for a detailed proof. While the maps ι∗ and p∗ are the ob-vious functorial maps, we define the connecting homomorphism ∂ = ∂n+1 : πn+1(Y, y0) −→πn(F, x0). Let α = [γ] ∈ πn+1(Y, y0), represented by a map γ : (In+1, ∂In+1) −→ (Y, y0).Let γ0 : In × 0 −→ X be the constant map x0. Then, we have the commutative diagram:

In × 0γ0 //

X

p

In × I γ

//g

;;

Y

2.4. CONSTRUCTION OF FIBRATIONS 49

where g is obtained by homotopy/left lifting property of the fibration p. It follows,pg(∂In) ⊆ γ(∂In+1) = y0. Therefore g(∂In) ⊆ F . Define g1 : In −→ F by g1(t) = g(t, 1).Then, g1(∂In) = x0. Therefore, g1 : (In, ∂In) −→ (F, x0). Define ∂(α) = [g1] ∈ πn(F, x0).

One checks that ∂ is well defined and the exactness.

2.4 Construction of Fibrations

This section is some excerpts from [W, Secton I.7]. Not surprisingly, we have some interestin the long exact sequence (2.5), while it does not appear to be easy to detect whether acontinuous map p : X −→ Y is a fibration or not. Given a continuous map p : X −→ Y ,we would replace it by a fibration X −→ Y where X is homotopic to X .

1. As usual I = [0, 1].

2. For a space Y , let P(Y ) := P(I, Y ) := F(I, Y ) be the space of all paths in Y . See(2.1.10) for topology on the function space F(I, Y ).

3. Suppose p : X −→ B be a continuous map.

4. LetIp = (x, u) ∈ X ×P(B) : p(x) = u(0) . So, Ip ⊆ X ×P(Y ).

More explicitly, Ip is the set pairs (x, γ) where x ∈ X and γ is part in Y , starting atp(x).

The following are some results from [W, Secton I.7].

Theorem 2.4.1. For any two spaces , X, Y the projections X × Y −→ X is a (White)fibration.

Corollary 2.4.2. The following are (White) fibration.p : P(X) −→ X ×X p(u) = (u(0), u(1))

p0 : P(X) −→ X p0(u) = u(0)

p1 : P(X) −→ X p1(u) = u(1)

Theorem 2.4.3. Composition of (White) fibration is a (White) fibration.

Theorem 2.4.4 (Hurewicz). Let p : X −→ B be continuous and B is paracompact. Then,p is a (White) fibration if there is a an open cover B = ∪iVi such that p−1Vi −→ Vi arefibrations.


Theorem 2.4.5. Let p : X −→ B be a fibration (any kind) and f : B′ −→ B be continuous.Then, the pullback p′ : X ′ −→ B′ is a fibration:

X ′ //

p′

X

B′

f// B

In particular, for A ⊆ B, the map p−1A −→ A is a fibration.

Proof. Easy, diagram chase.

Corollary 2.4.6. Let f : X −→ Y be a map. Then, If −→ X is a (white) fibration.

Proof. The following

Iff ′ //

p′0

P(Y )

p0

X

f// Y

is a pullback.

Since p0 is a fibration, so is p′0. The proof is complete.

We define a few functions:

For y ∈ Y ey : I −→ Y the constant map yε : Y −→ P(Y ) ε(y) = eyp0 : P(Y ) −→ Y p0(γ) = γ(0)p1 : P(Y ) −→ Y p(γ) = γ(1)f ′ : If −→ P(Y ) f ′(x, γ) = γp′0 : If −→ X p′0(x, γ) = xλ : X −→ If λ(x) = (x, εf)p : If −→ Y p = p1f

′

Following are helpful commutative diagrams:

Xf //

εf ""

Y

ε

P(Y )

So, εf(x) = ef(x).

X

1

εf

$$

λ

If

f ′ //

p′0

P(Y )

p0

X

f// Y

Iff ′ //

p""

p0

P(Y )

p1

X Y

Note, the map f : X −→ Y does not fit in the bottom of the last diagram.

2.4. CONSTRUCTION OF FIBRATIONS 51

Theorem 2.4.7. With the notations as above, we have

1. The map p = p1f′ : If −→ Y is a (White) fibration.

2. First„ p′0λ = 1X , λp′0 ' 1If

pλ = f, fp′0 ' p

3. So, importantly, p′0 and λ are homotopy equivalences.

4. Explicitly,

p((x, γ)) = p1f′(x, γ) = p1(γ) = γ(1) the other end of γ.

Proof. See [W, Thereom 7.30]. First, we prove that p = p1f′ is a (white) fibration.

Consider the commutative diagram:

Z × 0g //

_

If

p

Z × I

H//// Y

Write g(z, 0) = (g1(z), g2(z)(s)) ∈ If ⊆ X ×P(Y ) with s ∈ I.

Since the path g2(z)(s)) starts at f(g1(z)) and because of the commutativity, we have

g2(z)(0) = f(g1(z)) g2(z)(1) = H(z, 0)

For (z, t) ∈ Z × I, define a path

γ(z, t) : I −→ Y γ(z, t)(s) =

g2(z)(s) if t = 0g2(z)(s/t) if t 6= 0, s ≤ t

H(z, (s−t)t

1−t

)if 0 < t < s

Then, γ(z, t) is a path from g2(z)(0) = f(gi(z)). Also, γ(z, t)(1) = H(z, t). Now, define

G : Z × I −→ If G(z, t) = (g1(z), γ(z, t))

So, H lifts to a homotopy G. This establishes that f ′ is a (white) fibration.(It looks correct. But γ(z, t) looks like a loop: γ(z, t)(0) = γ(z, t)(1))

The equalities in (2) are obvious. The last equivalence follows from third and second:pλ = f =⇒ pλp′0 = fp′0 =⇒ p ' fp′0.

So, we need to prove λp′0 ' 1X . Note λp′0(x, γ) = (x, eγ(0)). Now,

With F ((x, γ(s)), t) = (x, γ(st))

F (−, 0) = λp′0F (−, 1) = 1

This completes the proof of (2). Now, (3) follows from (2).


Corollary 2.4.8. Suppose ∗ := y0 ∈ Y is the base point. Then, fiber

Tf := p−1(∗) = (x, γ) ∈ X ×P(Y ) : γ(1) = ∗ = y0, f(x) = γ(0)

So, (x, γ) ∈ Tf , if γ is a path from f(x) 7→ y0.

Following is the summary:

Corollary 2.4.9. Let f : X −→ Y be a continuous map. Then, we have the diagram

Tf // If

p′0 o

p // Y

Xf// Y

commute up to homotopy

and p is a White Fibration and hence a weak fibration. The fiber of p is Tf

So, up to homotopy, f fits in a Homotopy Fibration, which is the top line.

It follows from Lemma 2.3.4, there is a long exact sequence

· · · // πn(Tf , x0) // πn(X, x0) // πn(Y, y0)

∂n // πn−1(Tf , x0) // πn−1(X, x0) // πn−1(Y, y0)

// · · · · · · · · ·

// π1(Tf , x0) // π1(X, x0) // π1(Y, y0)

// π0(Tf , x0) // π0(X, x0) // π0(Y, y0)

(2.6)

Notation 2.4.10. For a map f : X −→ Y and y0 ∈ Y , Quillen [Q], uses the notationF (f, y0) := Tf , as described above (2.4.8). We would possibly, use Quillen’s notationssubsequently.

The Loop Space

Definition 2.4.11. With X = ∗ the singleton and f : X −→ Y , with f(∗) = y0, inCorollary 2.4.9, we write

ΩY := Tf to be called the loop space of Y at y0.

2.5. CW COMPLEXES 53

The loop space ΩY is to be considered as a pointed space, with constant path y0 to be thebase point. It follows form the long exact sequence (2.4.9) that

πi (ΩY, ∗) = πi+1 (Y, y0) ∀ i ≥ 0

2.5 CW Complexes

An very important category, associated to Top, is the category of CW complexes. Quillen’swork [Q] is heavily dependent on the topology of CW complexes. This material is availablein any standard source, and among them is the book of Hatcher [Ha]. Before we give adefinition of CW complexes, we give some definitions. As before (see Section 2.2), Dn ⊆ Rn

will denote the unit disc, ∂Dn := Sn−1 ⊆ Dn will denote the unit sphere, which is theboundary of Dn.

Definition 2.5.1. Let (X,A) be a topological pair and f : X −→ Y be a continuous map.Define the topological space X

∐f Y by the push forward diagram

A //

f

X

ιX

Y ιY// X∐

f Y

in Top.

So, X∐

f Y is constructed, set theoretically, from the disjoint union X∐Y by identifying

a ∼ f(a) ∀ ∈ A. The topology of X∐

f Y , is the largest one that makes ιX and ιY

continuous. We say that X∐

f Y is obtained from Y , by attaching X, along A, via f .

Of our particular interest would be the case, when (X,A) = (Dn,Sn−1), which encap-sulates the concept of "attaching cells".

Definition 2.5.2. Let n ≥ 0 and Un = x ∈ Dn :‖ x ‖< 1, denote the interior ofDn. Suppose X is a topological space. An n-cell of X is a subspace U ⊆ X, whichis homeomorphic of Un. A cell in X is an n-cell, for some n ≥ 0. An n-cell U would,sometimes, be denoted by Un, with the superscript, indicating the dimension. We alsowrite dim(U) = n, for an n-cell U .

A CW complex is defined as follow.

Definition 2.5.3. A CW complex is a topological space X, together with a sequence ofsubspaces X0 ⊆ X1 ⊆ · · · ⊆ Xn · · · ⊆ X, such that, the following conditions hold:

1. X = ∪∞i=0Xn.


2. X0 has discrete topology.

3. Inductively, Xn is obtained from Xn−1, by attaching n-cells enα, via maps ϕα :

Sn−1 −→ Xn−1. This means, there is an indexing set In and a family of contin-uous maps ϕα : Sn−1 −→ Xn−1 : α ∈ In,

ϕα : Sn−1 −→ Xn−1 : α ∈ In 3

∐nα∈I Sn−1 _

∐ϕα // Xn−1

∐nα∈I Dn // Xn

is a push forward in Top.

By definition (2.5.1), the topology on Xn is determined by this diagram.

4. A subset A ⊆ X is open (or closed) if and only if A ∩Xn is open (or closed) in Xn,for all n ≥ 0. (This condition is redundant, if X = Xn for some n.)

Following are comments and Information:

1. The above definition was taken from the book of Hatcher [Ha]. As is pointed out in[Ha], a variant of this definition in available in [W]. Many other definitions averrablein the literature, assumes that X is Hausdorff.

2. We have,

Xn =∐

α∈Im,m≤n

emα , X =∐

α∈Im,m≥0

emα are disjoint union.

The subspace Xn is called the n-skeleton of X.

3. For α ∈ In, the map ϕα extends as follows:

Sn−1 _

α // Xn−1

Dn //

Φα**

∐β∈In Dn // Xn

_

X

The map Φα is called the characteristic map of enα. The restriction (Φα)|Un : Un ∼−→enα is a homeomorphism.

4. It follows, a subset A ⊆ X is is open (or closed) if and only if Φ−1α (A) is s open (or

closed) in Dn, ∀ α ∈⋃∞n=0 In. To prove this, given a subset A ⊆ X, one needs to

check, A ∩Xn is open (or closed), which is done by induction.


5. For each n0 ≥ 0, Xn0 is closed in X. To see this, first note that, for any α ∈ In,Φ−1α (Xn−1) = Sn−1, is closed in Dn. Therefore, Xn−1 → Xn is closed in Xn, which

follows from pushout topology on Xn. Therefor Xn0 ∩Xn is closed in Xn, for all n.

6. Important Remark: More formally, one should say that, for given a topologicalspace X, a CW structure on X is given by the above process. Such a process providesthe above information, including the families ϕα : Sn−1 −→ Xn−1 : α ∈ In, n ≥ 0.

Alternately, a CW structure on X can be given by

(a) A family E := eα ⊆ X : α ∈ In, n ≥ 0 of cells on X such that

X =∐

α∈Im,m≥0

emα and, let Xn :=∐

α∈Im,m≤n

emα are disjoint union.

(b) For α ∈ In, a Characteristic maps Φα : Dn −→ X, such that (Φα)|Un : Un ∼−→ eα

is a homeomorphism and Φα(Sn−1) ⊆ Xn−1.

The push forward condition (3) would be a consequence of the above two condi-tions. Further, some authors like say (X,E ) is a CW complex and E is called a celldecomposition of X.

So, a topological space can have more than one CW structures.

We define sub complexes of CW complexes.

Definition 2.5.4. Suppose X is a CW complex, with cell decomposition E := eα ⊆ X :

α ∈ In, n ≥ 0. A closed subset Y ⊆ X is called a sub complex of X, if Y is union of asubfamily of cells aα : α ∈ J n, n ≥ 0, where J n ⊆ In.

The following is a key lemma on CW complex.

Lemma 2.5.5. Suppose X is CW complex. Let Y ⊆ X be a compact subset. Then, Y iscontained in a finite sub complex of X.

Proof. First, it is shown that Y can only meet finitely many cells in X. Inductively, fori = 1, 2, . . ., we can choose xi ∈ eniαi ∩ Y , where αi 6= αj ∀ i 6= j. Let Z = x1, x2 . . . , . Weclaim that Z is closed. Inductively, assume Z ∩ Xn−1 is closed. It would suffice to show∀ α ∈ In, Φ−1

α (Xn ∩ Z) is closed in Dn. Now,

Φ−1α (Xn ∩ Z) = Φ−1

α (Xn−1 ∩ Z) ∪ Φ−1α (enα ∩ Z)

By induction, the first one is a closed set in Sn−1 and the second one is a singleton or null.This establishes that Z is a closed set, and hence compact. Same argument shows that


any subset of Z is closed. Hence, Z is a discrete compact set. Therefore, Z is a finite set.This is a contradiction.

Therefore, Y meets only finitely many cells of X. In other words, Y is contained infinitely many cells. So, it is shown that Finite union of cells is is contained in a finite subcomplex. So, it is enough to show, each enα is contained in a finite sub complex.

Clearly, the zero cells e0α ⊆ X0 is closed in X0. Since X0 is closed in X, e0

α isclosed in X. So, zero cells are contained in a finite sub complex. Now, use induction.Consider the characteristic map Φα : Dn −→ X. Then, Φα(Dn) = ϕα(Sn−1) ∪ enα. Now,ϕα(Sn−1) ⊆ Xn−1. Therefore ϕα(Sn−1) ⊆ ∪ ∪mi=1 e

niαi, with ni ≤ n − 1. So, by induction,

ϕα(Sn−1) ⊆ A, where A ⊆ Xn−1 is a finite sub complex. Now, it is checked that A ∪ enα isclosed, because (A∪enα)∩Xn−1 is closed in Xn−1 and Φ−1

α (A∪enα) = Dn. For β ∈ ∪m≤nIm,α 6= β, Φβ(Sn−1) ⊆ Xn−1. Therefore, Φ−1

β (A ∪ enα) = Φ−1β (A), which is closed, in Xn and

hence in X. Therefore, A ∪ eα is a sub complex.


We proceed to prove that CW complexes are Hausdorff.

Lemma 2.5.6. Suppose X is a CW complex and x0 ∈ X. Then, x0 is closed.

Proof. Suppose x0 ∈ Xn \Xn−1. Then, x0 ∩Xn−1 = φ, and x0 ∈ enα, for some uniqueα ∈ In. Now, it follows from push out topology on Xn that x0 is closed in Xn and hencein X.

The following is an useful construction of open sets.

Construction 2.5.7. Suppose X is a CW complex, as above.

1. Let U ⊆ Xn0 be an open subset, in Xn0 . Then, there are open sets N(U) ⊂ X (notunique) such that N(U) ∩ Xn0 = U . The proof below would provide more explicitconstruction of such open sets.

2. Let U, V ⊆ Xn0 be open subsets, in Xn0 , with U ∩ V = φ. Then, there are open setsN(U), N(V ) ⊂ X such thatN(U)∩Xn0 = U , N(V )∩Xn0 = V andN(U)∩N(V ) = φ.

Proof. First, we give a easy version. For n ≥ n0, define Nn(U). Suppose, Nn(U) ⊆ Xn

one in Xn has been defined. For α ∈ In+1 define

Uα = Φα

(rx : x ∈ Φ−1

α Nn(U) ⊆ Sn, 0 < r ≤ 1)

Let Nn+1(U) = Nn(U) ∪

( ⋃α∈In+1

Uα

), which in open in Xn+1.

Let N(U) =⋃n≥n0

Nn+1(U), which in open in X.


This method can be refined, as follows. For all α ∈ ∪n≥n0+1In, let 0 < εα < 1. Redefine

Uα = Φα

(rx : x ∈ Φ−1

α Nn(U) ⊆ Sn, εα < r ≤ 1)

Then, N(U) defined as above, would be an open set in X.

Now, for U, V ⊆ Xn0 , the constructions above shows N(U) ∩N(V ) = φ. The proof iscomplete.

Proposition 2.5.8. Suppose X is a CW complex. Then, X is Hausdorff.

Proof. Let x 6= y ∈ X. Then, x ∈ emα and , y ∈ enβ. We assume m ≤ n. If n = m, wecan find open sets U, V , in Xn, with x ∈ U , y ∈ V and U ∩ V = φ (whether or not α = β.Let N(U), N(V ), be the open sets, as in (2.5.7). Then, N(U), N(V ) satisfy the Hausdorffconditions that x ∈ N(U), y ∈ N(V ) and N(U) ∩N(V ) = φ.

Now suppose m ≤ n− 1. Let U be an open neighborhood of x, in Xm. For k ≤ n− 1,construct Nn(U), as in (2.5.7). Let y0 ∈ Un such that Φβ(y0) = y. Let ‖ y0 ‖< εβ < 1.Now, define

U ′ := Uβ = Φβ

(rz : z ∈ Φ−1

β Nn(U) ⊆ Sn, εβ < r ≤ 1), V := Vβ = Φβ (z ∈ Dn :‖ z ‖< εβ)

Now, Uβ ∩ Vβ = φ and they are open neighborhood of x, y (respectively) in Xn. Now,N(U ′), N(V ) satisfy the Hausdorff property.

Proposition 2.5.9. Let X be CW complex. Then X is compactly generated.

Proof. We already proved that X is Hausdorff. Let A ⊆ X be a subset such that A ∩ Cis closed, for all compact subsets C of X. For α ∈ In, Φ(Dn) is compact. By hypothesis,A′ := A ∩ Φα(Dn) is closed. Therefore,

Φ−1α (A) = Φ−1

α (A′) is closed.

Therefore, A is closed. The proof is complete.

2.5.1 Product of CW Complexes

Let (X,E ), (Y,F ) be two CW complexes. Then, it is desirable that the topological spaceX×Y would have a natural CW structure. Unfortunately, this is does not a simple answer.We build up our thinking:

Denote the indexing set of E and In, n ≥ 0 and let Φα : Dn −→ X be the characteristicmaps. Denote the indexing set of F and J n, n ≥ 0 and let Ψβ : Dn −→ Y be thecharacteristic maps. Also,

E = enα : α ∈ ∪n≥0 In , F =εnβ : β ∈ ∪n≥0 J n


1. It may be more intuitive, to identify Dn ∼= In and Sn−1 ∼= ∂In. So, a characteristicmap would be considered as Φα : In −→ X or Ψβ : In −→ Y .

2. For α ∈ Im, β ∈ J n, there are the natural maps Φα × Ψβ : Im+n −→ X × Y . Note,Φα ×Ψβ is a homeomorphism on ∂Im+n.

3. Let (I ? J )n = ∪i+j=nI i × J j, I ? J = ∪n≥0 (I ? J )n and

E ?F =eiα × ε

jβ : (α, β) ∈ I ? J

The following theorem on product of CW complexes would be useful in Quillen K-Theory.

Theorem 2.5.10. Let (X,E ), (Y,F ) be two CW complexes. Use the notations Φα, Ψβ

for the characteristic maps, and other notations as above. Then, E ? F provides a CWstructure on k(X × Y ), the compactly generated topology corresponding to X × Y . If Xor Y is locally compact, then k(X × Y ) = X × Y .

Proof. See [Ha]. For the purpose of this proof, we denote Z := k(X × Y ). By definition,

Zn :=⋃

(α,β)∈I?J )m,m≤n

eiα × εjβ

Denote(Φ ?Ψ)n =

∐(α,β)∈(I?J )n

Φα ×Ψβ :∐

(α,β)∈(I?J )n

In −→ Zn

Note, the restriction of (Φ ?Ψ)n to ∂In maps in to Zn−1. One needs to prove that∐(α,β)∈I?J )n

∂In

// Zn−1

∐(α,β)∈I?J )n

In(Φ?Ψ)n

// Zn

is a pushout.

This obvious, set theoretically. Rest of the question is topological. The CW topology onX × Y is built, from bottom up, while topology on Z = k(X × Y ) is given. We need tocheck that these two topologies are same.

1. Suppose A ⊆ X×Y be closed in the Z = k(X×Y ). For each (α, β) ∈ (I ?J )n n ≥ 0the map Φα×Ψβ : In −→ X ×Y is continuous, with the product topology. We haveA∩ (Φα ×Ψβ(In)) is closed in the product topology. So, Φα ×Ψ−1

β (A) is closed. So,A is closed in the CW topology.

2. Suppose A ⊆ X×Y be closed in the CW topology. Suppose C ⊆ X×Y be a compactset. Then, C ⊆ C1×C2, where C1 ⊆ X, C2 ⊆ Y be projections of C. Then, C1 and C2

are compact. So, both C1 and C2 are, respectively, contained in finite sub complexesX1 = ∪r1i=1e

niαi⊆ X, Y1 = ∪r2j=1ε

miβj⊆ Y . It follows, C ⊆ X1×Y1 =

⋃r1i=1

⋃r2j=1 e

niαi×εmiβj .


To prove A∩C is closed in X×Y , we can replace X by X1 and Y by Y1 and assumeX, Y are finite. It follows

X1 × Y1 =

r1⋃i=1

r2⋃j=1

eniαi × εmiβj

=

r1⋃i=1

r2⋃j=1

Φαi ×Ψβj(In)

A∩(X1×Y1) =

r1⋃i=1

r2⋃j=1

Φαi×Ψβj(In)∩(A∩(X1×Y1)) =

r1⋃i=1

r2⋃j=1

Φαi×Ψβj

(Φαi ×Ψ−1

βj(A))

which is closed in X1 × Y1 ⊆ X × Y . Hence A ∩ C = A ∩ (X1 × Y1) ∩ C is closed inX × Y .

So, it is established that two topologies are same.

The latter part of the theorem is about product of compactly generated topologies.The proof is complete.

2.5.2 Frequently Used Results

First we define weak equivalences.

Definition 2.5.11. Suppose f : X −→ Y is a continuous map. We say f is a weakequivalence, if

∀ x0 ∈ X, ∀ n ≥ 0 πn(f) : πn (X, x0) −→ πn (Y, f(x0)) is a bijection.

Weak equivalence is an important concept, because of the following theorem of White-head.

Theorem 2.5.12. Let X, Y be two connected CW complexes. Then, a continuous mapf : X −→ Y is a weak equivalence, if and only if, f is a homotopy equivalence.

Proof. See [Ha, Theorem 4.5].

The following property of CW complexes is another tool used in [Q].

Proposition 2.5.13. Suppose X is a CW complex and Xn denotes the n-skeleton andx0 ∈ X0. Then,

the map πi (Xn, x0)

∼−→ πi (X, x0) is an isomorphism ∀ i ≤ n− 1

the map πn (Xn, x0) πn (X, x0) is surjective

Proof. See [Ha, Corollary 4.12].


Chapter 3

Simplicial and coSimplicial Sets

3.1 Introduction

These notes on Simplicial Sets is based on notes available on the Internet or from [H].

3.2 Simplicial Sets

Definition 3.2.1. For integers n ≥ 0, denote the set [n] := 0, 1, 2, . . . , n. Let ∆ be thecategory, whose objects are

[n] : n = 0, 1, 2, . . .

and morphisms are order preserving maps. So,

Mor∆([m], [n]) := ∆([m], [n]) := f : [m] −→ [n] : f(i) ≤ f(j) ∀ 0 ≤ i ≤ j ≤ m

This category is sometimes referred to as the Simplicial category.

1. ∆ has two obvious subcategories ∆+,∆− whose objects are same as that of ∆.The morphisms of ∆+ are consists of order preserving injective maps.The morphisms of ∆− are consists of order preserving surjective maps.

2. For integers n ≥ 1, and 0 ≤ i ≤ n, define special type of morphisms in ∆+

di : [n− 1] −→ [n] : di(k) =

k if k < i

k + 1 if i ≤ k ≤ n− 1”skip i”.

61

62 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Also, for integers n ≥ 1, and 0 ≤ i ≤ n− 1, define special type of morphisms in ∆−

si : [n] −→ [n− 1] : si(k) =

k if k ≤ i

k − 1 if i < k ≤ n”identifies i, i+ 1”.

3. cosimplicial identities:djdi = didj−1 ∀i < j

sjdi = disj−1 ∀i < j

= id i = j, j + 1

= di−1sj ∀ i > j + 1

sjsi = si−1sj ∀i > j

(3.1)

The following is an useful lemma.

Lemma 3.2.2. Let f : [n] −→ [m] in ∆ be a morphism in ∆. Then, f can be written(uniquely) as:

f = dikdik−1 · · · di1sj1sj2 · · · sjl

with0 ≤ j1 < j2 < · · · < jk < n 0 ≤ i1 < i2 < · · · < ik ≤ m

Proof. See [Mc]

Definition 3.2.3. Given a category C , a Simplicial Object in C a contravariant functor∆ −→ C . Likewise, a covariant functor ∆ −→ C is called a coSimplicial Object in C

We are mainly interested in simplicial sets, which are contravariant functors K :

∆ −→ Set. Such a simplicial set K, is given by

1. A sequence of sets Kn := K[n]. An element of x ∈ Kn is called an n-simplex and anelement x0 ∈ K0 is called a vertex.

2. The dual to di above, we have the face map di : Kn −→ Kn−1,

3. The dual to si above, we have the degeneracy map si : Kn−1 −→ Kn,

4. These maps are subject to the simplicial identities

didj = dj−1di ∀i < j

disj = sj−1di ∀i < j

= id i = j, j + 1

= sjdi−1 ∀ i > j + 1

sisj = sjsi−1 ∀i > j

(3.2)

These identities are obtained by reversing the arrows in (3.1).

3.2. SIMPLICIAL SETS 63

5. A map of simplicial sets τ : K −→ L is a natural transformation.Such a map is equivalent to a collection of maps τn : Kn −→ Ln commuting with theface and degeneracy maps:

Kn

τ

Kn−1oo sioo

τ

Kndi //

τ

Kn−1

τ

Ln Ln−1oo

sioo Ln di

// Ln−1

6. Suppose K a simplicial set K and x ∈ Kn is a simplex.In this case, we say dimx = n

Any image of x under arbitrary iterations of face maps dis is called a face of x.Images of x under iterated degeneracies are called degeneracies of x.Since 0-iteration is included, x is both a face and degeneracy of itself.A simplex x ∈ Kn is called non degenerate, if it is a degeneracy of itself ONLY.Given any simplex x ∈ Km, it is the degeneracy of a unique non degenerate simplexy ∈ Kn, for some n ≤ m. So, any x ∈ K0 is non degenerate, but its image s(x) ∈ K1

is not.

7. The simplicial sets, together with the maps of simplicial sets form a category, to bedenoted by SSet.

8. A map L −→ K of simplicial sets would be called a sub-simplical set, if Ln −→ Kn

is one to one, for all n ≥ 0. In other words, a sub-simplicial set is a simplicial set Lof K, such that Ln ⊆ Kn and the map L −→ K is a map of simplical sets. Further,given sub-simplicial set L ⊆ K, the quotient K

Lis define as follows:(

K

L

)n

=Kn

Ln.

Example 3.2.4. First, ∀ (fixed) n, the representable functor

∆(−, [n]) : ∆ −→ Sets given by ∆[n]([k]) = ∆([k], [n])

is a simplicial set. This is denoted by ∆[n] := ∆(−, [n]) ∈ SSet. So,

1. For each integer k ≥ 0, we can denote

∆[n]([k]) = (0 ≤ n1 ≤ n2 ≤ · · · ≤ nk ≤ n)


2. Non degenerate zero simplexes are given by ∆[n]([0]) are given by all the choicesof integers 0, 1, . . . , n. A non degenerate k simplex is given by a strictly increasingsequence 0 ≤ n1 < n2 < · · · < nk ≤ n.

Example 3.2.5. Then, ∀ (fixed) n, the boundary ∂∆[n] is defined to the smallest subsimplex of ∆[n] , whose not degenerate k simplexes are those of ∆[n], except the nondegenerate n-simplex ιn : [n]

∼−→ [n].

Example 3.2.6. ∀0 ≤ r ≤ n there is a simplicial set Λr[n] ∈ SSet, to be called the r-hornof ∆[n], whose nondegenerate k-simplices

= ι ∈ ∆([k], [n]) : ι is injective, ι 6= 1n, ι 6= dr

wheredr = (0, 1, . . . , r − 1, r + 1, . . . , n) : [n− 1] −→ [n]

is the face opposite of the vertex r.

Example 3.2.7. For n ≥ 0, we define Sn := ∆[n]∂∆[n]

∈ SSet. It has two non degeneratesimplexes ∗ ∈ Sn0 and σn ∈ Snn . (In deed, this quotient would correspond to the n-sphere,and so would ∂∆[n] and (∆[n], ∂∆[n])).

Reversing the arrows, one can also consider coSimplicial sets, which are the covariantfunctors K : ∆ −→ Set. Such a coSimplicial Set is given by

1. For each integer n ≥ 0, a set Kn.

2. For each f : [m] −→ [n] in ∆, (i. e. and order preserving map), a map K(f) :Km −→ Kn, satisfying obvious laws of compositions.

Perhaps, the most natural example of coSimplicial set (in fact, topological space) may bethe following.

Example 3.2.8. Let x0, x1, x2, . . . be a sequence of district symbols. For integers n ≥ 0,define

Σn =

n∑i=0

λixi : ∀ i 0 ≤ λi ≤ 1 andn∑i=0

λi = 1,

For any map f : [m] −→ [n] in ∆, define

Σ(f) : Σm −→ Σn by Σ(f)

(m∑i=0

λixi

)=

n∑k=0

∑f(i)=k

λi

xk

3.3. GEOMETRIC REALIZATION 65

3.3 Geometric Realization

Given a simplical set K, we would define its geometric realization |K|. In order to do that,we first define the geometric realization of ∆[n] as follows:

Example 3.3.1. We work in the category Top of topological spaces.

1. Let |∆[n]| ⊆ Rn be the convex hull of 0, e1, . . . , en with 0 = (0, . . . , 0) and e1, . . . , en

is the standard basis of Rn. So,

|∆[n]| :=

(t1, . . . , tn) ∈ Rn : 0 ≤ ti ≤ 1,∑

ti ≤ 1

∼=

(t0, t1, . . . , tn) ∈ Rn+1 : 0 ≤ ti ≤ 1,∑

ti = 1

2. An order preserving map f : [m] −→ [n] induces a map |∆[m]| −→ |∆[n]| sendingei 7→ ef(i).For the maps di ∈ ∆([n− 1], [n]), si ∈ ∆([n], [n− 1]), denote the corresponding mapsdi : |∆[n− 1]| −→ |∆[n]|, si : |∆[n]| −→ |∆[n− 1]|.

3. Hence [n] 7→ |∆[n]| is a cosimplicial topological space, to be denoted by |∆[−]|. Wewrite |∆[−]| ∈ Top∆. In fact, |∆[−]| is precisely the coSimplicial set described inExample 3.2.8

4. The topological space |∆[n]| is to be treated as the Geometric Realization of thesimplicial set ∆[n]. This is used to give a geometric realization |K| of any simplicialSet K ∈ SSet. The method given in [H], is a little too formal. So, we will use somewhat is available in the Net.

Definition 3.3.2. LetK ∈ SSet. Give eachKn the discrete topology. For f ∈ ∆([m], [n]),we have two maps

K(f) : Kn −→ Km and |f | : |∆[m]| −→ |∆[n]|.

1. For (τ, y) ∈ Km × |∆[m]| and (σ, x) ∈ Kn × |∆[n]|, define

(σ, x) ∼ (τ, y) if ∃ f ∈ ∆([m], [n]) 3 K(f)(σ) = τ, |f |(y) = x.

2. We use all three notations, (σ, x) ∼ (τ, y), (σ, x) ∼f (τ, y) or (σ, x) ∼f (τ, y), in theabove case. We would also say the f induces this relation. The relationship ∼ isreflexive and transitive, but not necessarily symmetric.


3. Let ' denote the equivalence relation generated by ∼, on∐∞

n=0Kn × |∆[n]|. Note,this equivalence relation can be described, using the face and degeneracy maps only.

Define the geometric realization

|K| :=∐∞

n=0Kn × |∆[n]|'

where∐

denotes the disjoint union. (3.3)

Possible choices of topology on the geometric realization |K| are as follows:

1. The geometric realization |K| inherits a topology from Kn × |∆[n]|. We would referto this topology by the "usual topology".

2. The more frequently used, and standard, topology on |K| comes from the CW struc-ture on |K|, described subsequently.

Lemma 3.3.3. This association | − | (3.3.2), defines a functor

| − | : SSet −→ Top

Conversely, define the following functor:

Definition 3.3.4. Define

S : Top −→ SSet by S(Y )n = MorTop (|∆[n]|, Y )

This will be denoted by Sing in [H, Rem 3.1.7].

Lemma 3.3.5. These two functors

| − | : SSet −→ Top and S : Top −→ SSet

are adjoint to each other:

Φ : MorTop (|K|, Y )∼−→MorSSet (K,S(Y )) ∀ K ∈ OB(SSet), Y ∈ OB(Top).

Outline of the Proof. For objects K in SSet and Y in Top, we define the bijectionsof the adjoint map: Given f : |K| −→ Y and an integer n ≥ 0, we obtain the diagonalcompositions map in Top, as follows:

Kn × |∆[n]| //

&&

|K|fY

This gives a map Φ(f)n : Kn −→MorTop(|∆[n]|, Y ) = S(Y )n


Define

Φ : MorTop (|K|, Y )∼−→MorSSet (K,S(Y )) by Φ(f) = Φ(f)n : n ≥ 0.

This way we get a map, subject to checking further details:

MorTop (|K|, Y ) −→MorSSet (K,S(Y ))

Conversely, given F : K −→ S(Y ), we have

Fn : Kn −→ S(Y )n = MorTop (|∆[n]|, Y )

This defines a map Fn : Kn × |∆[n]| −→ Y Top.

One checks that Fn factors through a continuous map Ψ(F ) : |K| −→ Y . Now, Ψ definesa maps

Ψ : MorSSet (K,S(Y )) −→MorTop (|K|, Y )

One further checks that Φ and Ψ are inverse of each other and Φ satisfies other conditionsof Adjoints. The proof is complete.


3.3.1 The CW Structure on |K|

Given a simplicial set K ∈ SSet, we would provide a CW structure on |K|. First, wedefine the skeletons of |K|, as follows.

Definition 3.3.6. Suppose K ∈ SSet.

1. For integers n ≥ 0, let Kn ⊆ K be the subsimplicial set generated by the set ∪ni=0Ki.

2. We would establish that |Kn| defines a CW structure on |K|.

We prove the following lemma.

Lemma 3.3.7. Let K ∈ SSet and n ≥ 1 be an integer. Suppose (σ, x), (σ′, x′) ∈ Kn−1 ×|∆[n− 1]| and (τ, y) ∈ Kn × |∆[n]|, such that

(σ, x) ∼ (τ, y) ∼ (σ′, x′) in |Kn| 3 ” ∼ ” are induced by a face or degeneracy.

Then, either τ ∈ Kn−1 or there is a chain of equivalences, of length two,

(σ, x) ∼ (τ ′, y′) ∼ (σ′, x′) where (τ ′, y′) ∈ Kn−2 × |∆[n− 2]|

and the equivalences are induced by face or degeneracy maps.

Proof. We consider various cases:

1. Suppose both relations are induced by face maps. Let

di, dj ∈ ∆([n− 1], [n]) be faces, and

(σ, x) ∼di (τ, y)

(σ′, x′) ∼dj (τ, y)

In this case, K(di)τ = σ |di|(x) = yK(dj)τ = σ′ |dj|(x′) = y

Degeneracies in ∆([n−1], [n−2]) would be denoted by ζ i and faces in ∆([n−1], [n−2])would be denoted by δi. If i = j, then σ = σ′ and x = x′. So, we assume i < j. Wehave

K(djδi) = K(δi)K(dj) = K(δj−1)K(di) = K(diδj−1)

It follows, K(δj−1)K(di)τ = K(δj−1)(σ) |di|(x) = yK(δi)K(dj)τ = K(δi)(σ′) |dj|(x′) = y

With y = (y0, y1, . . . , yn) we have yi = yj = 0


Let x0 = |ζ isj|(y) = |ζj−1si|(y) = (y0, . . . , yi−1, yi+1, . . . , yj−1, yj+1, . . . , yn)and σ0 = K(δj−1)(σ) = K(δi)(σ

′)

We claim(σ0, x0) = (K(δj−1)(σ), x0) ' (σ, x) given by δj−1 : [n− 2] −→ [n− 1].(σ0, x0) = (K(δi)(σ′), x0) ' (σ′, x′) given by δi : [n− 2] −→ [n− 1].

One checks,|di|(|δj−1|(x0)) = |di|(x) = y =⇒ |δj−1|(x0) = x.|dj|(|δi|(x0)) = |dj|(x′) = y =⇒ |δi|(x0) = x′.

So, the claim is established and (σ, x) ∼ δj(σ0, x0) ∼δi (σ′, x′).

2. Now suppose, one of the two equivalences is induced by a degeneracy. Assume(σ, x) ∼si (τ, y). In this case, τ = K(sj)(σ) ∈ Kn−1.

This completes the proof.

Proposition 3.3.8. Suppose K ∈ SSet For integers m ≤ n, consider the commutativediagram of geometric realizations:

|Km| ιmn // q

ιm ""

|Kn| _ιn|K|

For all integers m ≤ n, the maps ιmn, ιn, ιn are injective (as the notations indicate).

Proof. It is enough to prove, for integers n ≥ 1, the map ι := ι(n−1)n : |Kn−1| −→ |Kn| isinjective. Let ω, ω′ ∈ |Kn−1|, with ι(ω) = ι(ω′). We can write ω = [(σ, x)], ω′ = [(σ′, x′)],for some (σ, x), (σ′, x′) ∈ Kn−1 × |∆[n− 1]|. There a chain of equivalences

(σ, x) = (τ0, y0) ∼µ1 (τ1, y1) ∼µ2 · · · ∼µr (τr, yr) = (τ, y)

where µi are face or degeneracy maps and (τi, yi) ∈ Kni × |∆[ni]|. We use induction ofr. For r = 2, it follows from Lemma 3.3.7, that ω = ω′ in |Kn−1|. If τi ∈ Kn−1, forsome i = 1, . . . , r − 1, then by induction, it follows ω = ω′ in |Kn−1|. So, we assume thatni ≥ n for all i = 1, . . . , r − 1. Let N = maxni : i = 1, . . . , r − 1. Now, again, we useinduction of N . If N = n, by repeated application of Lemma 3.3.7, it follows that ω = ω′

in |Kn−1|. Now, assume N ≥ N − 1. Again, by Lemma 3.3.7, for i = 1, . . . , r − 1, witheither τi ∈ KN−1 or can be replaced by a (τ ′i , y

′i) ∈ KN−2 × |∆[N − 2]|. Since the "height"

N has reduces, we have ω = ω′ in |Kn−1|. The proof is complete.


The CW Structure on |K|We proceed to describe the CW structure on |K|. Subsequently, ∂ |∆[n]| would denote

the boundary of l |∆[n]|. Further,

Interior(|∆[n]|) = (t0, t1, . . . , tn) : 0 < ti < 1 ∀ i,∑

ti = 1

1. Let, |K0| = K0, be endowed with discrete topology. For integers n ≥ 0, write

Kn =∐k≤n

Kk × |∆[k]|

Then, the map Kn −→ |Kn| is surjective, sending (σ, x) 7→ [(σ, x)].

2. From construction of the Geometric representation (Equation 3.3), for each σn ∈ Kn,there is a map

ασn : |∆[n]| ∼−→ σn × |∆[n]| −→ |Kn|Now, suppose σn is non degenerate.

(a) For m ≤ n− 1 and

(σm, x) ∈ Km×|∆[m]|, (σn, y) ∈ σn×|∆[n]|, (σm, x) ∼ (σn, y) =⇒ y ∈ ∂|∆[n]|

(b) It follows from the proof of Proposition 3.3.8, the map

ασn : |∆[n]| ∼−→ σn × |∆[n]| −→ |Kn| is injective on Interior(|∆[n]|)

(c) On the other hand,

∀ x ∈ ∂ |∆[n]| ασn(x) ∈ Image(Kn−1).

For example,

ασn(0, t1, t2, . . . , tn) = [(σn, (0, t1, t2, . . . , tn))] = [(d0σn, (t1, . . . , tn))]

(d) Note, |∂∆[n]| = ∂ |∆[n]|. Also note, ασn(|∂∆[n]|) ⊆ |Kn−1|. Denote

∂(ασn) := (ασn)|∂|∆[n]| : |∂∆[n]| −→∣∣Kn−1

∣∣3. Let K ′n ⊆ Kn be the set of all non degenerate simplexes. Let

D :=∐

σ∈K′ |∆[n]|, and α :=∐

σ∈K′ ασ : D −→ |Kn|S :=

∐σ∈K′ |∂∆[n]|, and ∂(α) :=

∐σ∈K′ ∂ (ασ) : S −→ |Kn−1|

Proposition 3.3.9. LetK be a simplicial set. With the notations, as above, for all integersn ≥ 1, the following

S ∂(α) // _

i

|Kn−1| _ι(n−1)n

D α

// |Kn|

is a push forward diagram in Set.


Proof. We use the universal property of push forward. Extend the above to the diagram:

S ∂(α) // _

i

|Kn−1| _ι(n−1)n

f

D α//

g,,

|Kn|ϕ

""Z

In the diagram, f, g are given, such that that outer diagram commute. We need to provethat there is a unique map ϕ, such the outer triangles commute. When exists, uniquenessof ϕ follows, because |Kn| is union of the images of α and ι(n−1)n. For (σ, x) ∈ Km×|∆[m]|,with m ≤ n and ω = [(σ, x)], defne

ϕ(ω) =

f(ω) if m ≤ n− 1g(x) if m = n

Here x is considered in the σ−componenent of D.

One checks that ϕ is well defined. The proof is complete.

Lemma 3.3.10. Suppose K ∈ SSet. Then, the geometric realization |K| has a CWcomplex structure. In this CW structure |Kn| would be the n-skeleton. The n-cells aregiven, restrictions of the map ασ to the interior of |∆[n]|.

Proof. Follows from Proposition 3.3.9.

The following is a basic result on geometric realization.

Proposition 3.3.11. Suppose K is a simplicial set and L ⊆ K is a simplical subset. Then,the map |L| −→ |K| is injective. Further, |L| is a closed subset of |K|

Proof. For a non degenerate n-simplex σ ∈ K, let CellK(σ) ⊆ |K| denote the corre-sponding n-cell in |K|. In addition to the notation Kn ⊆ K, as above, let Ln denote thesimplicial subset of L, generated by simplifies of dimension ≤ n. Note, for integers n ≥ 0there is commutative diagram of natural maps

|Ln| κn //

ι′n

|Kn|ιn

|L| κ// |K|

Let K ′n denote the nondegenrate n-simplices of K and L′n denote the nondegenrate n-simplices of L. For σ ∈ L′n, if σ = K(si)τ for τ ∈ Kn−1, then τ = K(di)σ ∈ L would


be a degeneracy in L. Hence, L′n ⊆ K ′n. Also, note for σ ∈ L′n, |∆[n]| ∼−→ CellL(σ)∼−→

CellK(σ) are homeomorphism. Now, we have

|L| =∐

σ∈∪n≥0L′n

CellL(σ) ⊆∐

σ∈∪n≥0K′n

CellK(σ) = |K| ,

where∐

denotes disjoint union.

To see |L| is a closed subset of |K|, we will prove integers n ≥ 0, |Ln| ⊆ |Kn| is aclosed subset. Clearly, |L0| −→ |K0| is closed. Now, consider the commutative diagram ofpushout diagrams:

Sn−1L q

""

// _

|Ln−1| _

r

κn−1

$$Sn−1K

// _

|Kn−1| _

DnL αL

// q

""

|Ln| rκn

$$DnK αK

// |Kn|

where

DnK =

∐α∈K′n

Dn Sn−1K = ∂Dn

K

DnL =

∐α∈L′n

Dn Sn−1L = ∂Dn

L

By induction |Ln−1| is closed in |Kn−1|, hence in |Kn|. Also, α−1K (|Ln|) = Dn

L is closed inDnK . So, |Ln| is closed in |Kn|. The proof is complete.

3.4. THE HOMOTOPY GROUPS OF SIMPLICIAL SETS 73

3.4 The Homotopy Groups of Simplicial Sets

In this section, we define the homotopy groups of simplicial sets K ∈ SSet. Easiest way todefine homotopy groups πn (K, v0) is to define it as that of the geometric realization |K•|.

Definition 3.4.1. Suppose K ∈ SSet. Note, for all n ≥ 0, ∆[0]n are singletons and|∆[0]| = ? is a point.

1. Let v0 ∈ K0 be a vertex. Then, v0 determines a point in |K|, which is the equivalenceclass of (v0, ?) ∈ K0 × |∆[0]|. We denote this point as |v0| ∈ |K|.

2. The map [n] −→ [0] induce the map (iterated degeneracy) K0 −→ Kn. Given avertex v0 ∈ K0, it defines a sub simplicial set V (v0) ⊆ K, where V (v0)n has a singlesimplex, namely, the image of v0 under the map K0 −→ Kn.

By abuse of notations, V (v0) is, sometimes denoted by v0 ⊆ K. So, |v0| ⊆ |K| is the"same point", defined above.

Define, nth-homotopy group

πn (K, v0) := πn (|K| , |v0|) (Compare [H,Prop 3.6.3])

However, there has been attempts to define these homotopy groups/sets combinatorially,with some success. We extract (in Section 3.4.1) the combinatorial definition of πn (K, v0)

from [H], when K is "Fibrant". Although we do this, it would be of little help for ourpurposes, in the context of K-Theory, which we would clarify latter.

3.4.1 The Combinatorial Definition

First, we define product of two simplicial sets.

Definition 3.4.2. Let X, Y ∈ SSet. Define X × Y ∈ SSet by

∀ n ∈ N (X × Y )n := Xn × Yn

Given f ∈ ∆([m], [n] define

(X × Y )(f) := X(f)× Y (f) : Xn × Yn −→ Xm × Ym

In particular, the face and degeneracy maps are given by product of the same.

Remark. In fact, in the category Top, |X ×Y | ∼−→ |X| × |Y | is a homeomorphism, usualtopology is considered. However, as is remarked after the proof of [H, Lemma 3.1.8], |X×Y |and |X| × |Y | are not necessarily homeomorphic, with compactly generated topology.


Now we define homotopy of simplicial sets. Denote by I := ∆[1] ∈ SSet. So,

I0 = ∆([0], [1]) = 0, 1, I1 = ∆([1], [1]) = (0, 0); (0, 1), (1, 1)

I2 = ∆([2], [1]) = (0, 0, 0), (0, 0, 1); (0, 1, 1); (1, 1, 1) · · ·Now, 0, 1 ∈ I0 induces a sub simplicial sets 0,1 ⊆ I, also viewed as 0,1 : ∆[0] −→ I = ∆[1].

Definition 3.4.3. Suppose X, Y ∈ SSet and f, g : X −→ Y are two maps in SSet.We say that f is homotopic to g, if

∃ F : X × I −→ Y in SSet 3 F (x, 0) = f(x), F (x, 1) = g(x) ∀ x ∈ Xn n ∈ N

where F (−, 0) and F (−, 1) are defined as follows:

X∼ //

F (−,0)**

X ×∆[0]1X×0// X × I

FY

and

X ∼ //

F (−,1)**

X ×∆[0]1X×1// X × I

FY

Further, suppose A ⊆ X and B ⊆ Y be subsimplicial sets. A maps, f, g : (X,A) −→(Y,B) are said to be homotopic, if there is a homotopy

F : X × I −→ Y 3

F (−, 0) = f

F (−, 1) = g

F (A× I) ⊆ B

It is of our particular interest, when both A,B represent vertices. In that case, we say f, gare base point preserving maps and H is a base point preserving homotopy.

Definition 3.4.4. A simplicial setX ∈ SSet, is said to satisfy the Kan extension conditionif ∀ n ∈ N, 0 ≤ k ≤ n and any map f : Λk[n] −→ X in SSet, extends to a map ∆[n] −→ X

in SSet. Diagramatically,

Λk[n]f //

_

X

∆[n]

∃ϕ

==

Such a simplicial set X is called FIBRANT.

More generally, a map γ : X −→ Y is called a fibration, if γ has the right liftingproperty, as in the diagram:

Λk[n]f //

_

X

γ

∆[n]

∃ϕ

==

F// Y


Meaning, if the outer diagram of map commute, there in a map ϕ so that the inner trianglescommute.

In particular, a simplicial set X is fibrant ⇐⇒ the (unique) map X −→ ∆[0] is afibration.

Example 3.4.5. Note ∆[1] does not satisfy Kan extension condition.

Λ0[2]f //

_

∆[1]

∆[2]

f(0, 1) = (0, 1)

f(0, 2) = (0, 0)

Digress: Non degenerate simplices (which we denote with ′)(∆[2])′0 = (0), (1), (2)(∆[2])′1 = (0, 1), (1, 2), (0, 2)(∆[2])′2 = (0, 1, 2)

(Λ0[2])′0 = (0), (1), (2)(Λ0[2])′1 = (0, 1), (0, 2)

So, f is well defined because rest is forced by simplicial identities. In deed, it follows,

f(0) = f(d1(0, 1)) = d1(f(0, 1)) = d1(0, 1) = 0, Likewise, f(1) = 1, f(2) = 0.

Now, suppose there is an extension F : ∆[2] −→ ∆[1], of f . So, see this let F (0, 1, 2) =

(i, j, k) with 0 ≤ i ≤ j ≤ k ≤ 1. Then,F (1, 2) = F (d0(0, 1, 2)) = d0(i, j, k) = (j, k)

1 = F (1) = F (d1(1, 2)) = d1(F (1, 2)) = d1(j, k) = j

0 = F (2) = F (d0(1, 2)) = d0(F (1, 2) = d0(j, k) = k

This contradicts that j ≤ k.

Example 3.4.6. Recall the adjoint functors | − | : SSet −→ Top, S : Top −→ SSet.Now, for Y ∈ OB(Top), SY has Kan extension property.

Proof. Consider the diagram

Λk[n]f //

_

SY

∆[n]

By Adjunction

MorTop(|K|, Y ) ∼= MorSSet(K,SY ) =⇒MorTop(|Λk[n]|, Y ) ∼= MorSSet(Λk[n], SY )


Corresponding to f , there is a continuous map f : |Λk[n]| −→ Y . I put it in the diagram

∣∣Λk[n]∣∣ f //

_

Y

|∆[n]|

Also, ∃ retract r : |∆[n]| −→∣∣Λk[n]

∣∣

Hence, f r : |∆[n]| −→ Y . By Adjunction, there is F corresponding to f r, completing thecommutative diagram

Λk[n]f //

_

SY

∆[n]

∃ F

<<


Under certain conditions, homotopy is an equivalence relation. To prove this the fol-lowing Lemma would be useful.

Lemma 3.4.7. Let X, Y ∈ SSet and Y satisfies the Kan extension condition. Then amap of simplicial sets f : X × Λk[n] −→ Y extends to a map F : X × ∆k[n] −→ Y .Diagramatically:

X × Λk[n]

f // Y

X ×∆[n]

F

::

Proof. Skip!

Now we prove the equivalence.

Lemma 3.4.8. Let X, Y ∈ SSet and Y satisfies the Kan extension condition. Then thehomotopy (') is an equivalence relation on the set of all simplical maps X −→ Y . Sameis true for base point preserving maps.

Proof. Let f, g, h : X −→ Y be maps of simplicial maps.

1. (Reflexivity): Consider the diagram

X × I π //

F ##

X

fY

where π is the projection, and F = fπ.

Then, F (0) = F (1) = f .


2. (Symmetry.) Suppose F : X × I −→ Y be a homotopy, with F (0) = f, F (1) = g.Recall, the non degenerate simplexes:

I ′ = ∆[1]′ =

0, 1(0, 1)

→ ∆[2]′ =

0, 1, 2(0, 1), (0, 2), (1, 2)(0, 1, 2)

Λ1[2]′ =

0, 1, 2(0, 1), (1, 2)

For i < j, the sub simplicial set generated by the nondegenrate (i, j), will be denotedby [(i, j)]. Suppose F : X × I −→ Y be a homotopy: F (0) = f, F (1) = g. Define

F ′ : X × Λ1[2] −→ Y

F ′X×I = FF ′(x, [(1, 2)]) = g(x)

This is well defined, because F ′(x, 1) = g = F (1), at the vertex 1, where two legsmeet. By Lemma 3.4.8, F ′ extends to G′ : X×∆[2] −→ Y . Let G : X×[(0, 2)] −→ Ybe the restriction of G′. Then, G(0) = f,G(2) = g.

3. (Transitivity.) Consider homotopiesF : X × [(0, 1)] −→ YG : X × [(1, 2)] −→ Y

3F (0) = f F (1) = gG(1) = g G(2) = h

In deed, F,G defines a simplicial map F ′ : X × Λ1[2] −→ Y . By Lemma 3.4.8, F ′extends to a simplicial map G′ : X × ∆[2] −→ Y . Now, G′(0) = f and G′(2) = h.So, f ' h.


3.4.2 The Definitions of Homotopy Groups

We give two definitions of homotopy sets πn(−,−). The following is the first definition.

Definition 3.4.9. For integer n ≥ 0, let 0 ∈ ∆[n]0 be the base point. Let (X, x0) be apointed simplicial set (i., e x0 is a vertex of X. Then, define

πn (X, x0) = Set of all homotopy equivalence class of maps f : (∂∆[n], 0) −→ (X, x0)

The definition from the book of Hovey [H]

Definition 3.4.10. Suppose X ∈ SSet is a fibrant simplicial set. For x, y ∈ X0 definex ∼ y (homotopic), if there is z ∈ X1 such that d1z = x, d0z = y.

Lemma 3.4.11. Suppose X is a fibrant simplicial set. Then homotopy of vertices is anequivalence relation. We denote the set of equivalence classes by π0X.


Proof. We have:

1. (Reflexivity): Let x ∈ X0. Recall

∀ i = 0, 1

[0] di //

1

[1]

s0

[0]

=⇒X0

s0 //

1 !!

X1

diX0

=⇒ d0(s0(x)) = x = d1(s0(x))

2. (Symmetry): Suppose x ∼ y ∈ X0. So, ∃ z ∈ X1 3 d1z = x, d0z = y. Recall,

∆[1]′ =

0, 1(0, 1)

∆[2]′ =

0, 1, 2d2 := (0, 1), d1 := (0, 2), d0 := (1, 2)(0, 1, 2)

Λ0[2]′ =

0, 1, 2d1, d2

(a) Note s0x, z ∈ X1.

(b) Nondegenerate simplexes of Λ0[2] (that count) are d1, d2 ∈ ∆([1], [2]).

(c) Define f : Λ0[2] −→ X

d1 7→ s0xd2 7→ z

(d) Diagramatically, represent ∆[2] by i2 := 1[2] = (0, 1, 2):

∆[2] :

•1

d0i20•

d2i2

d1i2

2 •

and Λ0[2] :

•1

0•

d2i2

d1i2

2 •

The map f : Λ0[2] −→ X IS

•1

0•

d2i2

d1i2

2 •

7→

•y

x•

z

s0x

•x

(e) Since X is fibrant, f extends to F : ∆[2] −→ X:

•1

d0i20•

d2i2

d1i2

2 •

7→

•y

wx•

z

s0x

•x

=⇒w = d0F (i2) = F (d0(i2)) = F (1, 2)d1w = y, d2w = x.


3. (Transitivity): Suppose x ∼ y ∼ z. By symmetry, we assume z ∼ y.So, ∃ a, b ∈ X1, 3 d1a = x, d0a = y = d1b, d0b = z:

(a) Define f : Λ0[2] −→ X

d1 7→ bd2 7→ a

. Diagramatically:

•1

0•

d2i2

d1i2

2 •

7→

•x

y•

a

b

•z

(b) By fibrant property, f extends to:

•1

d0i20•

d2i2

d1i2

2 •

7→

•x

cy•

a

b

•z

=⇒ d1(c) = x, d0c = z.

Lemma 3.4.12. Suppose X is a fibrant simplicial set. Then there is a natural isomorphismπ0X ∼= π0|X|.

Outline of the Proof. For a vertex x ∈ X0, let [v] ∈ π0X denote the class of v. Recall,π0|X| is the set of all path components of |X|. Define a map

ϕ : π0X −→ π0|X| by sending [v] 7→ v × ? ∈ π0|X| =∐

nKn × |∆[n]|∼

One checks that this map is well defined and and a bijection.

Definition 3.4.13. With notations as above, for any fibrant simplycial set X and anyvertex

v ∈ X0, define π0(X, v) := (π0X, [v]) as a pointed set.

Higher Homotopy Sets πn(X, v):

Now, we proceed to define πn(X, v), for all integers n ≥ 0. Before that, we introducethe following definition and notations.


Definition 3.4.14. Suppose X, Y are two simplicial sets. Define a new simplicial setMap(X, Y ) as follows:

1. For integers n ≥ 0, let Map(X, Y )n := MorSSet (X ×∆[n], Y ).

2. For a morphism f : [m] −→ [n] in the category ∆, let Map(f) : Map(X, Y )n −→Map(X, Y )m be the map induced by ∆(f) : ∆[m] −→ ∆[n].

So, Map(X, Y ) has the structure os a simplicial set.

Further, we comment, that given a vertex v ∈ Y0, v defines a vertex in Map(X, Y ),that means an element in Map(X, Y )0 = MorSSet(X, Y ), which we continue to denote byv, as follows:

X //

v!!

∆[0]

v

Y

This is the constant map.

Of our subsequent interest would be the vertex in Map(∆[n], Y ), for all integers n ≥ 0,defined by the vertex v in Y :

∆[n] //

v##

∆[0]

v

Y

Definition 3.4.15. Let X ∈ SSet be fibrant and v ∈ X0 be a vertex. Note that therestriction map Map(∆[n], X) −→ Map(∂∆[n], X) is also a map of simplicial sets. Let Fbe the fiber (pullback) of this map, over the vertex v:

F

//Map(∆[n], X)

v //Map(∂∆[n], X)

By [H, Thm. 3.3.1], the second vertical map is a fibration. Hence F is fibrant. Define

πn(X, v) := π0(F, v) = (π0(F ), [v])

The following is some dissections:

1. We have Map(∆[n], X)0 = MorSSet(∆[n], X) = Xn,Map(∂∆[n], X)0 = MorSSet(∂∆[n], X),Map(∆[n], X)1 = MorSSet(∆[n]×∆[1], X)Map(∂∆[n], X)1 = MorSSet(∂∆[n]×∆[1], X)


2. So, F0 consists of all α ∈ Xn, all whose faces are v. Equivalently,

all α : ∆[n] −→ X such that α(∂∆[n]) = v.

3. To define π0(F ), recall for α, β ∈ F0 one defines α ∼ β if

∃ H ∈ F1 3 d0(H) = α, d1(H) = β.

This means, there is a map of simplicial sets

H : ∆[n]×∆[1] −→ X 3 H(−, 0) = α,H(−, 1) = β,H|∂∆[n]×∆[1] = v.

Remark: There should be a direct proof, that the above is an equivalence relation onF0. If and when we had that, we be able to avoid using the result [H, Thm. 3.3.1]that Map(∆[n], X) −→Map(∂∆[n], X) is a fibration, if X is a fibrant.

4. Here is an important correspondence:

Lemma 3.4.16. Let X ∈ SSet be fibrant and v ∈ X0 be a vertex. Then,

πn(X, v) = πn(|X|, |v|) ∀ n ≥ 0.

Therefore, πn(X, v) is a group for n ≥ 1 and is abelian for n ≥ 2.

Outline of the Proof. A complete proof is given in [H, Prop. 3.6.3]. The casen = 0 was proved before (3.4.12). Fix an integer n ≥ 0. Let α ∈ F0. That meansα : ∆[n] −→ X such that α|∂∆[n] = v. It follows, the geometric realization of α is amap |α| : (|∆[n]|, |∂∆[n]|) −→ (|X|, |v|). Define

ϕ : (X, v) −→ πn(|X|, |v|) := MorTopPair (|∆[n]|, |∂∆[n], (|X|, |v|)) by ϕ([α]) = [|α|]

It follows easily, by geometric realization of the homotopies, that ϕ is well defined.Now, one checks that ϕ is bijective.

5. Suppose f : X −→ Y is a map of fibrant X, Y ∈ SSet and v ∈ X0 is a vertex. Then,f induces a map of the homotopy groups πn(X, v) −→ πn(Y, f(v)), making thesegroups functorial:

π0 : SSet −→ Setπ1 : SSet −→ Grπn : SSet −→ Ab n ≥ 2

The Connecting Homomorphisms

Definition 3.4.17. Let p : X −→ Y be a fibration of fibrant simplical sets X, Y andv ∈ X0 be a vertex. Let [α] ∈ πn (Y, p(v)). We have lift of α as follows:

Λn[n] v //

X

p

∆[n]

γ

==

α// Y

=⇒∆[n− 1]

dn

dnγ // X

p

∆[n]

γ

;;

α// Y


By definition pdnγ = dnpγ = dnα = p(v). Let F be the fiber of p over p(v):

∆[n− 1]

%%

dnγ

&&

F //

X

p

∆[0]

p(v)// Y

So, dnγ lies in F . Further, didnγ = v, by looking at Λn[n].So, dnγ ∈ πn(F, v). This defines a map

∂ : πn (Y, p(v)) −→ πn−1 (F, v) .

Lemma 3.4.18. Let p : X −→ Y be a fibration of fibrant simplicial sets X, Y and v bea vertex of X. Let i : F −→ X be the fiber of p over p(v). Then, there is a sequence ofpointed sets

· · · ∂ // πn (F, v) // πn (X, v) // πn (Y, p(v))

∂ // πn−1 (F, v) // πn−1 (X, v) // πn−1 (Y, p(v))

· · · · · · · · ·

∂ // π1 (F, v) // π1 (X, v) // π1 (Y, p(v))

∂ // π0 (F, v) // π0 (X, v) // π0 (Y, p(v))

This sequence is exact, (in the sense that the the kernel (the premiere of the base point)is equal to the image of at each degree.

Remark 3.4.19. We tried to give a flavor that Homotopy Theory of Simiplicial Sets can bedeveloped without any reference to Topological Homotopy Theory (as in [H, Chapter 3]).However, we refrained from going in to deeper details of the homotopy theory of SimplicialSets. For our purpose, Definition 3.4.1, should suffice, that for a simplicial set X (fibrantor not) and a vertex v of X:

πn(X, v) := πn(|X|, v) ∀ n ≥ 0

Quillen [Q], did not seem to have used the Homotopy Theory of Simplicial Sets. As heindicates [Q, pp. 5-6], such theory was not in existence, at that time. Even in [H, Chapter

3.5. FREQUENTLY USED LEMMMS 83

3], it was defined form Fibrant Simplicial Sets. In our context of Quillen K-Theory, wemay not find this combinatorial approach, because the simplicial set we consider (i. e.nerve of a category) may not be Fibrant.

3.5 Frequently Used Lemmms

Lemma 3.5.1. Let X ∈ SSet and ` ∈ X1 be an 1-simplex. Then, there is a path γ(`) :

I −→ |X|, such that γ(0) = d0(`) and γ(1) = d1(`).

Proof. Recall

|∆[0]| = ? and |∆[1]| = (t0, 1− t0) : 0 ≤ t0 ≤ 1

Define β : I −→ X1 × |∆[1]|, by β(t) = (`, (1− t), t). Let γ(`) be the composition:

Iβ //

γ(`) $$

X1 × |∆[1]|

|X|

By notations, d0(`) := (d0(`), ?) ∼ (`, (0, 1)) = γ(0). Likewise, d1(`) = γ(1). The proof iscomplete.

The following is about push forward and geometric realization of simplical sets.

Lemma 3.5.2. Suppose

AfK //

fL

K

ιK

L ιL// K∐

A L

is a push forward diagram in SSet or in SimTop.

Then, the diagram of the geometric realizations:

|A| |fK | //

|fL|

|K||ιK |

|L||ιL|// |K

∐A L|

is a push forward diagram in Top or in CGHaus.


Proof. (Double Check) Clearly, the diagram of geometric realization commutes. Considerthe commutative diagram diagram of continuous maps.

|A| |fK | //

|fL|

|K||ιK | α

|L||ιL|//

β ,,

|K∐

A L|

X

Define,

ϕ :

∣∣∣∣∣K∐A

L

∣∣∣∣∣ =(Kn

∐AnLn)× |∆[n]|∼

−→ X by ϕ(([σ, x)])

α(([σ, x)]) if σ ∈ Kn

β(([σ, x)]) if σ ∈ Ln

One checks that ϕ is well defined and continuous. It is clear that ϕ |ιK | = α and ϕ |ιL| = β.Uniqueness of such a maps ϕ is obvious.

3.5.1 A Lemma on Bisimplicial Sets

In this section, we discuss a lemma on Bisimplicial Sets. Before we proceed, we introducea nation.

Notation 3.5.3. The category of Simplicial Topological spaces would be denoted bySimTop. So, objects of SimTop, are functors

∆o −→ Top.

Given two objects, K,L : ∆o −→ Top,

Mor(K,L) = Set of all natural transformations τ : K −→ L

In particular, for such a τ : K −→ L, the maps τn : Kn −→ Ln are continuous maps.

Remark: First, not that there is a forgetful functor SimTop −→ SSet. On the otherhand, any set X with its discrete topology can be considered as a topological space. Inthis way, there is a natural functor SSet −→ SimTop.

Now, we define Bisimplicial Sets.

Definition 3.5.4. A Bisimplicial Set T is a functor

T : ∆o ×∆o −→ Set. For ([m], [n]) ∈ Obj(∆o ×∆o) write Tmn = T([m], [n]).


Similarly, a functor

T : ∆o ×∆o −→ Top. is called a BiSimplicial Space

Subsequently, in this section, we treat BiSimplical Sets as Bisimplicial Topological Spaces.

Given such a Bisimplicial space T, we define, essentially three Simplicial Spaces, asfollows:

1. First, the diagonal DT : ∆o −→ SimSet, is defined by (DT)n := (DT)([n]) := Tnn.Then, DT ∈ SimTop. We call it the diagonal of T.

2. For a fixed integer m ≥ 0, define Rm• : ∆o −→ SimSet, by Rm•([n]) = Tmn. So,RTm• ∈ SimSet. Likewise, for a fixed integer n ≥ 0, define LT•n : ∆o −→ Set, byLT•n([m]) = Tmn. So, LT•n ∈ SimSet.

3. Now, define, two functors

LT,RT : ∆o −→ Top by

(LT)m := (LT)[m] := |Rm•|(RT)n := (LT)[n] := |L•n|

Therefore, LT,RT ∈ SimTop are simplicial topological spaces. We will comparethe geometric realizations of DT,LT,RT ∈ SimTop, in the next lemma (3.5.5)

The following is in [Q, pp. 86].

Lemma 3.5.5. Let T : ∆o ×∆o −→ SimSet be a bisimplicial space and DT, LT, RT beas above (3.5.4). Then, there are natural homeomorphisms:

|LT| |DT|∼oo

∼// |RT|

Proof. See [Q]. We establish only the first homeomorphism. We have|DT| =

∐m≥0 Tmm×|∆[m]|

∼

|LT| =∐

m≥0|Rm•|×|∆[m]|∼ =

∐m≥0

(∐n≥0 Tmn×|∆[n]|

∼

)×|∆[m]|

∼

From this, if follows that there is a map

ϕ : |DT| −→ |LT| where ϕ([(τ, x)]) = [((τ, x)], x)] ∀ (τ, x) ∈ Tmm × |∆[m]|


Now, one proves that ϕ is a homeomorphism. This is done in two stages. For notationalconveniences, for integers r, s ≥ 0, we introduce the notation

Ω[r, s] := ∆[r]×∆[s] : ∆o ×∆o −→ Set. So,Ω[r, s]([p], [q]) = ∆([p], [r])×∆([q], [s]) ∀ p, q ≥ 0.

Clearly, Ω[r, s] is bisimplicial set. Further, for fixed topological space S and integers r, s ≥0, T be defined by

Tmn = Ω[r, s]× S = ∆([m], [r])×∆([n], [s])× S

In our familiar notations:

∆o ×∆0 Ω[r,s] //

T((

Set

×S

// Top

Top

We fist establish the Lemma for such a bisimplicial set T. We have

|DT| =∐

m≥0 ∆([m], [r])×∆([m], [s])× S × |∆[m]|∼

∼−→ S × |∆[r]| × |∆[s]|

Similarly, for a fixe integer m ≥ 0, we have

|RTm•| =∐

n≥0 ∆([m], [r])×∆([n], [s])× S × |∆[n]|∼

∼−→ ∆([m], [r])× S × |∆[s]|

Therefore,

|LT| =∐

m≥0 |RTm•| × |∆[m]|∼

=

∐m≥0 ∆([m], [r])× S × |∆[s]| × |∆[m]|

∼∼−→ S×|∆[r]|×|∆[s]|

This establishes that ϕ is a homeomorphism, in this case.

Now, establish the same for general Bisimplicial spaces T.

1. For integers r, s ≥ 0, there is a map (natural transformation) of bisimplicial sets,

εrs : Ω([r, s])×Trs = ∆[r]×∆[s]×Trs −→ T defined by εrs(f, g, τ) = T (f, g)(τ) ∈ Tmn,

where (f, g, τ) ∈ ∆([m], [r])×∆([n], [s])× Trs.

We note the following:

(a) εrs is a map of bisimplicial spaces.

(b) εrs is determined by the equation εrs(ιr, ιs, τ) = τ , where ιr denotes the nondegenerate simplex of ∆[r].

(c) Therefore, the image of εrs is the faces and degeneracies of elements in Trs.


2. Combining, we get a surjective map

ε :=∐

r≥0,s≥0

εrs :∐

r≥0,s≥0

Ω[r, s]× Trs −→ T

So, ε is surjective.

3. Given (f, g) ∈ ∆([r], [r′])×∆([s], [s′]) there are maps Tr′s′ −→ Trs and a map ∆[r]×∆[s] −→ ∆[r′]×∆[s′]. These two maps, induce maps of bisimplical spaces:

Ω[r, s]× Tr′s′ ∆[r]×∆[s]× Tr′s′(f,g,1) // ∆[r′]×∆[s′]× Tr′s′ Ω[r′, s′]× Tr′s′

∆[r]×∆[s]× Tr′s′(1,1,(f,g))

// ∆[r]×∆[s]× Trs Ω[r, s]× Trs

In fact, the diagram

Ω[r, s]× Tr′s′(f,g,1) //

(1,1,(f,g))

Ω[r′, s′]× Tr′s′εr′s′

Ω[r, s]× Trs εrs

// T

commutes.

4. By taking coproduct, we have a commutative diagram∐S Ω[r, s]× Tr′s′

∐(f,g,1)//

∐(1,1,(f,g))

∐(r′,s′) Ω[r′, s′]× Tr′s′

ε∐

(r,s) Ω[r, s]× Trs ε// T

(3.4)

wehre S = (f, g, r, r′, s, s′) : r, r′, s, s′ ≥ 0, (f, g) ∈ ∆([r], [r′])×∆([s], [s′]).One checks, that this is a push forward diagram of bisimplicial spaces. To check thisone needs to check that the for all integers, p, q ≥ 0, the (p, q)-component of theabove diagram is a push forward in Top:∐

(f,g)∈∆([r],[r′])×∆([s],[s′]) ∆([p], [r])×∆([q], [s])× Tr′s′∐

(f,g,1) //

∐(1,1,(f,g))

∐(r′,s′) ∆([p], [r′])×∆([q], [s′])× Tr′s′

ε′

∐(r,s) ∆([p], [r])×∆([q], [s])× Trs ε”

// Tpq

where ε′, ε” denote the restrictions of ε. In fact, both ε′, ε” are surjective maps. Itfollows form construction of push forward (in Set), that above rectangle is a pushforward diagram.Therefore, it is established the the diagram (3.4) is push forward diagram of bisimpli-cial sets. Hence, if follows the D of the diagram (3.4) are also push forward diagrams.Further it follows from Lemma 3.5.2 that and the L of the above diagram (3.4) arealso push forward diagrams.


Now, consider the commutative diagram

D (∐

S Ω[r, s]× Tr′s′)

∼ϕ

**

//

∐(r′,s′) D (Ω[r′, s′]× Tr′s′)

∼ϕ

++L (∐

S Ω[r, s]× Tr′s′) //

∐(r′,s′) L (Ω[r′, s′]× Tr′s′)

D(∐

(r,s) Ω[r, s]× Trs)∼ϕ

))

// DT

∼ϕ

**L(∐

(r,s) Ω[r, s]× Trs)

// LT

We need to prove that the lower ring hand map ϕ is a bijection. Since other three arebijection, by properties of push out, so is the fourth one. (In the notes, broken arrows havebeen uses to indicate existence of an arrow. However, here the natural maps ϕ was alreadydefined above.

Chapter 4

The Quillen-K-Theory

This chapter is essentially an exposition of the paper of Quillen [Q]. Other such expositionsinclude [Sv, Sm1, Wc2].

Before we proceed, we introduce the following:

Notations 4.0.6. Recall, a category C is called a small category, if Obj(C ) is a set.

1. The category of small categories and functors will be denoted by Cat.

We would often, work with categories C , such that there is an equivalence F : Cs∼−→

C where Cs is in Cat. Such a category, C is said to have a set of isomorphismclasses of objects.

2. The category of CW complexes would be denoted by CW. The category of pointedCW complexes would be denoted by CW•. Also, the category of pairs CW complexeswould be denoted by CWPair.

4.1 The Classifying spaces of a category

This section mostly corresponds to [Q, §1].

Definition 4.1.1. Suppose C is a small category. To any such category, we associate asimplicial set N C ∈ SSet, as follows:

1. We let, the set of vertices N C0, be the set of all objects in C . The set of n-cellsN Cn be the set, sequence of composable arrows

σ : X0f1 // X1

f2 // X2f3 // · · · // Xn−1

fn // Xn(4.1)

89

90 CHAPTER 4. THE QUILLEN-K-THEORY

2. The face maps di : N Cn −→ N Cn−1 is given by sending, the n-call σ in (4.1), bydropping Xi, and for i 6= 0, n,

replacing Xi−1fi−1 // Xi

fi // Xi+1by Xi−1

fifi−1 // Xi+1

3. The degeneracy maps si : N Cn −→ N Cn+1 is given by sending, the n-call σ in(4.1),

replacing Xi−1fi−1 // Xi

fi // Xi+1by Xi−1

fi−1 // Xi Xifi // Xi+1

The simplicial set N C ∈ SSet is called the nerve of C .

Definition 4.1.2. Suppose C is a category. The classifying space BC is defined to be thegeometric realization |N C | ∈ Top. So, BC := |N C |.

So, objects X ∈ Obj(C ) corresponds to a vertex (a point) in BC .

Remark: There is no reason, that the nerve N C is a category is a Fibrant simplicial set.

Following obvious lemma would be very useful.

Lemma 4.1.3. Suppose C is a category and f : X −→ Y is a morphism. Then there is aa path γ := γ(f) : [0, 1] −→ BC such that γ(f)(0) = X and γ(f)(1) = Y .

Proof. Follows from Lemma 3.5.1.

Following are two useful examples of classifying spaces.

Example 4.1.4. Let I denote the category of the partial ordered set 0, 1. Note, thenerve N (I) = ∆[1]. So, the classifying space BI = [0, 1], the interval.

Example 4.1.5. Suppose G is a group. Let G denote the category with one object ? andMor(?, ?) = G. Then,

πi (G, ?) =

G if i = 1

0 if i 6= 1

Here the isomorshism G∼−→ πi (G, ?) sends g ∈ G to the loop corresponding to g : ? −→ ?

(see Lemma 3.5.1).

Proof. Skip

4.1. THE CLASSIFYING SPACES OF A CATEGORY 91

4.1.1 Properties of the Classifying Space

Proposition 4.1.6. Let C ,D be two small categories and F : C −→ D be a functor.Then, F induces a continuous map B(F ) : BC −→ BD . This associations

C 7→ BC

F 7→ B(F )defines a functor B : Cat −→ Top.

Proof. Follows from construction.

Proposition 4.1.7. Let C be a small category. Then, there is a (cellular) homeomorphismBC

∼−→ BC o, where C o is the dual category.

Proof. Follows from construction.

Proposition 4.1.8. Let C ,D be two small categories.

1. It follows easily that N (C ×D)∼−→ N (C )×N (D) is a bijection.

2. So, there is a continuous bijection B(C ×D) −→ BC × BD . This map is a homeo-morphism, if the right side is given by compactly generated topology. (Also, if BC orBD is a finite complex.) In particular, if BC or BD is compact.

3. In particular B(C × I) ≡ BC × I, where I denotes the category (0 < 1) and I = [0, 1].(In fact, the nerve N I ∼= ∆[1].)

Proposition 4.1.9 (Prop. 2). Let C ,D be two small categories and F,G : C −→ D be twofunctors. Suppose θ : F −→ G is a natural transformation. Then, θ induces a homotopy

H : BC × I −→ BD 3 H(0) = B(F ), H(1) = B(G).

Proof. For a morphism f ∈MorC (X, Y ) we have the commutative diagram

F (X)F (f) //

θX

F (Y )

θY

G(X)G(f)

// G(Y )

Define a functor

H : C×I −→ D 3

H(X, 0) = F (X), H(X, 1) = G(X) ∀ X ∈ Obj(C )H(f, ι) = F (f) : F (X) −→ F (Y ) ∀ f ∈MorC (X, Y ), ι ∈MorI(0, 0)H(f, ι) = G(f) : G(X) −→ G(Y ) ∀ f ∈MorC (X, Y ), ι ∈MorI(1, 1)H(f, ι) = θY F (f) = G(f)θX ∀ f ∈MorC (X, Y ), ι ∈MorI(0, 1)


By (4.1.6, 4.1.8) there is a homotopy

H := H : BC × [0, 1] ≡ B(C × I) −→ BD .

If follows, H(−, 0) = BF and H(−, 1) = BG. The proof is complete.

Corollary 4.1.10. Let F : C −→ D , G : D −→ C be two functors of small categoriesand F be left adjoint to G. Then, B(F ) : BC −→ BD is a homotopy equivalence.

Proof. There is a natural transformation 1C −→ GF as follows:

MorD(FA,X)ηAX∼//MorC (A,GX) =⇒

MorD(FA, FA)

η

∼//MorC (A,GFA)

MorD(FGX,X) η∼ //MorC (GX,GX)

Let θ(A) = η(1FA) : A −→ GFA. Then, θ : 1C −→ GF is a natural transformation.Therefore 1BC ' B(G)B(F ) are homotopic. Likewise, B(F )B(G) ' 1BC are homotopic. So,B(F ) is a homotopy equivalence.

Corollary 4.1.11. Suppose C is a category with an initial object or a final object. Then,BC is contractible.

Proof. Assume C has an initial object 0. Let 0 denote the full subcategory of C consistingof the zero object. (This is the category with on object, one morphism.)

Let F : 0 −→ C be the inclusion functor. Let G : C −→ 0 be the functorG(X) = 0 ∀ X ∈ Obj(C ) and G(f) = 10 ∀ f ∈MorC (X, Y ). Then,

with A = 0 ∈ Obj(0), X ∈ ObjC

MorC (0, X) ∼ //Mor0(0, 0)

MorC (FA,X) Mor0(A,GX)

This shows that F is a left adjoint functor. Hence ∗ −→ BC is a homotopy equivalence,by (4.1.10).

4.1.2 Directed and Filtering Limit

Recall the following definition.

Definition 4.1.12. A partially ordered set (I,≤) is called a directed set, if ∀ i, j ∈ I thereis a k ∈ I such that i ≤ k, j ≤ k.

As small category I is called a filtering category, if


1. ∀ i, j ∈ Obj(I) there are two arrows:

if

k

j

g

@@ (4.2)

2. Given two arrows λ, β : i −→ k, there is an arrow γ : k −→ κ such that γλ = γβ.Diagramatically,

i λ

β//

γλ=γβ

k

γ

κ

It follows, a directed set is a filtering set.

Definition 4.1.13. Suppose I is a filtering small category. A functor A : I −→ Set, alsowritten as Ai : i ∈ I is called a filtering family of sets.Then, a functor C : I −→ Cat, also written as Ci : i ∈ I is called a filtering family ofsmall categories.

Proposition 4.1.14. Suppose I is a filtering category and Ci : i ∈ I is filtering familyof small categories, indexed by I.

1. Let C = lim→ Ci. (One should call them coLimits.)That means, C is a category, with functors Ci −→ C ∀ i ∈ I such that:

∀ i ≤ j

Ci

##C

ϕ // D

Cj

?? <<

the first triangle commute. Further, given the commutative diagram of the outertriangle, there is a unique functor, ϕ such that all the triangles commute.

Since we are working with small categories, this is essentially a limit in in the categoryof sets, as follows. In particular, such limits exist.


2. It follows,

Obj(C ) = lim→ I

Obj(Ci) =

⊔i∈I Obj(Ci)

∼.

Denote ιi : Obj(Ci) −→ Obj(C ).

MorC (X, Y ) = lim→ I,ιi(Xi)=X,ιi(Yi)=Y

MorCi(Xi, Yi)

3. It follows that, the nerve N (C ) = lim→ I N (Ci).

4. Let Xi ∈ Obj(Ci) be a family, such that X = ιi(Xi). Then, there are maps

πn(Ci, Xi) −→ πn(C , X).

Proposition 4.1.15 (Prop 3). With notations, as in (4.1.14), we have

lim→ I

πn(Ci, Xi) = πn(C , X) ∀ n ≥ 0.

Proof. We have N C = lim→ I N Ci. The rest of the proof follows from the correspondingresult of simplicial sets (4.1.16), below.

Proposition 4.1.16. Let I be a filtering category and Ki : i ∈ I be a filtering familyof simplicial sets. Let K = lim→ I K

i and v = lim→ I vi ∈ K, where vi ∈ Ki0 be vertices

∀ i ∈ I. Then,lim→ I

πn(|K|i, vi) = πn(|K|, v) ∀ n ≥ 0.

Outline of the Proof. Note, (|K|, v) = lim→(|Ki|, vi), in the Category CW•. The restof the proof follows, from the corresponding results on CW complexes, as follows (4.1.17)..

Proposition 4.1.17. Let I be a filtering category and X i : i ∈ I be a filtering familyin the category of CW complexes (in particular, the maps X i −→ Xj are in the categoryCW). . Let X = lim→X

i and v = lim→ I vi ∈ K, where vi ∈ Ki0 be vertices ∀ i ∈ I.

Then,

lim→ I

πn(X ivi) = πn(X, v) ∀ n ≥ 0.


Outline of the Proof. By usual arguments, we have the following commutative diagramsof the "limit" maps:

(X i, vi)

ιij

ιi

%%(Xj, vj) ιj

// (X, v)

which induce maps

πn (X i, vi)

ηi

πn(ιi)

((

((

lim→I πn (X i, vi) ι// πn (X, v)

Here, the first triangle is in the category of pointed CW complexes, and the second one isin Set. Using results on CW-structures, which we did not cover, the proof is completed asfollows:

πn(X, v) = MorTop• ((Sn, ?), (X, v)) = MorTop•

((Sn, ?), lim

→I(X i, vi)

)∼= lim→I

MorTop•

((Sn, ?), (X i, vi)

)= lim→I

πn(X i, v).

This argument depends on the facts that X i, i ∈ I is a filtered family in CW and thatSn is compact. The proof is complete.

Corollary 4.1.18. Suppose ∀i ≤ j, the map BCi −→ BCj is a homotopy equivalence.Then, BCi −→ BC is a homotopy equivalence.

Proof. Obvious!

Corollary 4.1.19. Suppose I is a filtering category. Then, I is contractible.

Proof. We need to prove that BI is contractible. In fact, I = lim→ I/i (meaning, BI/iis contractible.. Since, I/i has a final object, BI/i is contractible. By (4.1.18), BI iscontractible.

Definition 4.1.20. From now on, terminologies and concepts from Topology, would beborrowed freely to Category Theory. A category C would said to have a "topologicalproperty", if the classifying space BC has the same property. For example,

1. A functor F : C −→ D would be called a homotopy equivalence, if BC −→ BD is ahomotopy equivalence.

2. A category C would be called contractible, if BC is contractible.

3. Two functor F,G : C −→ D would be called a homotopic, if B(F ) and B(G) arehomotopic.


4.1.3 Theorem A: Sufficient condition for a functor to be homo-

topy equivalence

Definition 4.1.21. Suppose F : C −→ D is a functor and Y ∈ Obj(D).

1. Define the category Y \ F , as follows:Obj(Y \ F ) = (A, u) : A ∈ Obj(C ), u : Y −→ FA ∈MorD.For (A, u), (B, v) ∈ Obj(Y \ F ) a morphism is a morphism

w : A −→ B ∈MorC 3

FA

Fw

Y

u

==

v !!FB

commutes.

In particular, if f is identity then Y \ Id is a category under Y :

= (A, u) : u : Y −→ A

2. Likewise, for an object Y ∈ Obj(D), the category F/Y is defined as follows:Obj(F/Y ) = (A, u) : u : FA −→ Y ∈MorD.For (A, u), (B, v) ∈ Obj(F/Y ) a morphism is a morphism

w : A −→ B ∈MorC 3

FA

Fw

u

!!Y

FB

v

== commutes.

The following is the celebrated Theorem A in the paper of Quillen [Q].

Theorem 4.1.22 (Theorem A). Let F : C −→ D be a functor, such that Y \ F iscontractible, ∀ Y ∈ Obj(D). Then, F is a homotopy equivalence. Dually (see Prop 4.1.7),if F/Y is contractible, ∀ Y ∈ Obj(D). Then, F is a homotopy equivalence.

Proof. Let S(F ) be the category, described as follows:

1. The objects are

Obj(S(F )) = (X, Y, v) : X ∈ Obj(C ), Y ∈ Obj(D), v ∈MorD (Y, FX)


2. For (X, Y, v), (X ′, Y ′, v′) ∈ Obj(S(F )), Defined MorS(F ) ((X, Y, v), (X ′, Y ′, v′)) =(u,w) ∈MorC (X,X ′)×MorD(Y ′, Y ) :

Y ′ w //

v′

Y

v

FX ′ FXFuoo

commute

There are functors:

S(F )p1 //

p2

C

Do

3p1(X, Y, v) = X p1(u,w) = up2(X, Y, v) = Y p2(u,w) = w

Let T (F ) be the bisimplicial set, meaning the functor

∆o ×∆o −→ Sets

defined by

T (F )pq := T (F )(p, q)

=

Yp

fp // Yp−1// · · · // Y0

v0 // FX0 in D

X0 u1// X1 u2

// · · · // Xq−1 uq// Xq in C

For morphisms (f, g) in ∆o ×∆o, define T (F )(f, g) accordingly.

Treat the simplicial set N (C ) (the nerve), as a bisimplicial set, by (p, q) 7→ N (C )q.Now consider the map of bisimplicial sets:

T (F ) −→ N (C ) projections T (F )pq −→ N (C )pq = N (C )q (4.3)

Given an ordered pair in T (F )pp, as described above, let

vk = F (uk) · · ·F (u1)v0f1 · · · fk : Yk −→ FXk. So, (Xk, Yk, vk) ∈ S(F ), and

(X0, Y0, v0)(u1,f1)

// (X1, Y1, v1) // · · · // (Xp, Yp, vp) ∈ N (S(F ))p.

This shows that, the nerve N (S(F )) is the diagonal p 7→ T (F )pp (i.e. in bijection). Thegeometric realization of the projective p1, gives a map:

B(p1) : BS(F ) = |T (F )pp| −→ BC

Now, for a fixed q|p 7→ T (F )pq| =

∐X•∈N (C )q

B(D/FX0)o


Since D/FX0 has a final object, namely, 1 : FX0 −→ FX0, B(D/FX0) ' ? is contractible.So, there is a homotopy equivalence

|p 7→ T (F )pq| =∐

X•∈N (C )q

B(D/FX0)o −→∐

X•∈N (C )q

? = N Cq

is a homotopy equivalence, for all q.

Write Tq = |p 7→ T (F )pq| , So, T = Tq is a simplicial topological space.

It follows from above, ∀ q ≥ 0, the map

Tq −→ N (C )q is a homopoty equivalence,with discrete topology on N (C )q.

It follows from theorem of May and Tornehave (I did not check) or from [Q, Lemma] that

|T | = |(q 7→ |p 7→ T (F )pq|)| ' BC is a homotopy equivalence.

By Lemma 3.5.5 we have the commutative diagram

BS(F ) |T (F )pp|B(p1) //

o

BC

|T | |(q 7→ |p 7→ T (F )pq|)|

66

Here the vertical map is a homeomorphism, by Lemma 3.5.5 and the diagonal map is ahomotopy equivalence. Therefore, the horizontal map B(p1) is a homotopy equivalence.

Now we consider the functor p2 : S(F ) −→ Do. As before, consider N (Do)pq =N (Do)p, as a bisimplicial set and we have a map of bisimplicial sets. As in (4.3), considerthe projection map to the Y -coordinate:

T (F ) −→ N (Do) sending T (F )pq 7→ N (Do)p (4.4)

As before, one can check that T (F )pp = N (S(F )). So, the classifying map B(p2) is givenby

BS(F ) |T (F )qq|B(p2) // BDo

In fact, for any fixed Y• ∈ N (Do), we have

(x•, y•) ∈ T (F )pq : y• = Y• ' (Y0 \ F )

By hypothesis, there is a homotopy equivalence

B(Y0 \ F ) −→ pt

So, for fixed p, we have |q 7→ T (F )pq| =∐Yp // Yp−1

// · · · // Y0

B(Y0 \ F ) −→∐

Yp // Yp−1// · · · // Y0

pt = N (D0)p


which is homotopy equivalence, ∀ p.

BS(F ) |T (F )qq|B(p2) //

o

BDo

|(p 7→ |q 7→ T (F )pq|)|

66

The vertical map is a homeomorphism and the diagonal map is a homotopy equivalence.So, B(p2) is a homotopy equivalence.

Now, we have a commutative diagram

Do S(F )p1 //p2oo

f

C

F

Do S(id) π1//

π2oo D

This induces the corresponding map of the classifying spaces:

B (Do) B (S(F ))B(p1) //B(p2)oo

B(f)

B (C )

B(F )

B (Do) B (S(id))

B(π1)//

B(π2)oo B (D)

All the horizontal maps are homotopy equivalence. Hence B(f) is a homotopy equivalence.And, hence B(F ) homotopy equivalence.


A Corollary to Theorem A

Definition 4.1.23. Let F : C −→ D be a functor. For Y ∈ Obj(D), let

F−1(Y ) = X ∈ Obj(C ) : FX = Y → C denote the subcategory

Arrows of F−1(Y ) are those that map to 1Y .

1. First, we define prefibred category, via a functor F .

(a) We say that F makes C a prefibred category, if the functor

ι : F−1(Y ) −→ Y \ F given by X 7→ (X, 1X)

has a right adjoint.

(b) Given a prefibred category C , given by F : C −→ D , denote the right adjointby

ζ : Y \ F −→ F−1(Y ), and write ζ(X, v) = v∗X

(c) (Not Needed) Given a morphism v : Y −→ Y ′, we have the following triangleof functors:

F−1(Y ′) θ //

v∗ %%

Y \ Fζ

F−1(Y )

where

θ(X) = (X, v)

ζ(X, v) =: v∗X

This functor v∗ is determined up to canonical isomorphism (what does it mean?).This s called the Base Change.

(d) (Not Needed) A prefibred category C , given by F : C −→ D , is called a fibredcategory, if for all pair of compassable arrows Y u // Y ′

v // Y ” the canonicalmorphism of functors u∗v∗ −→ (vu)∗ is an isomorphism.

For X ∈ F−1(Y ), I do not see a natural map u∗v∗(X) −→ (vu)∗(X). I believeit comes from properties of adjoint functors.

2. Dually, we define precofibred category, via a functor F .

(a) We say, F makes C a precofibred category, if the functor

F−1(Y ) −→ F/Y has a left adjoint β (X, v : F (X) −→ Y ) 7→ v∗X


(b) (Not Needed) Now, suppose v : Y −→ Y ′ is a morphism in D . Then, we havethe commutative diagram of functors:

F−1(Y ) Θ //

v∗ %%

F/Y ′

β

F−1(Y ′)

Θ(X) = (X, v)

β(X, v) =: v∗(X)

We say, v∗ is cobase change.

(c) (Not Needed) We say, C is coffered category, if (vu)∗∼−→ v∗u∗.

Corollary 4.1.24. Suppose F : C −→ D is either prefibred or precofibred, such thatF−1(Y ) is contractible, for all objects Y in D. Then, F is a homotopy equivalence.

Proof. We prove the first case. By Corollary 4.1.10, F−1(Y ) −→ Y \ F is a homotopyequivalence. Since F−1(Y ) is contractible, so is Y \ F . Now, the corollary follows fromTheorem A (4.1.22).

4.1.4 Theorem B: The Exact Homotopy Sequence

In this section, we apply the long exact sequences of homotopy groups/set in topology(2.6), to Classifying spaces of categories and functors, to obtain a long exact sequence ofhomotopy groups/sets of categories. That would be the objective of Theorem B (4.1.32),which is proved by an application Theorem A (4.1.22).

Some Further Background From Topology:

Before we proceed, we bring in some definitions from topology.

Definition 4.1.25. Let g : E −→ B be a map of topological spaces and b ∈ B. The,Homotopy Fiber F(g, b) was defined in (2.4.8, 2.4.10). In the cartesian product notation,the Homotopy Fiber can be described, as follows:

F(g, b) = E ×B BI ×B b = (z, γ, b) ∈ E ×BI × b : γ ∈ BI , γ(1) = b, γ(0) = g(z)

This leads to a long exact sequence of homotopy groups/sets (2.6).

More generally, we recall the definition of homotopy cartesian product.


Definition 4.1.26. Consider the maps h, g of topological spaces

E

g

B′

h// B

and extend it to

E ′ h′ //

g′

E

g

B′

h// B

(4.5)

a commutative diagram of topological spaces. Write

E(g, h) := B′×BF(I, B)×BE := (b′, γ, e) ∈ B′ × F(I, B)× E : γ(1) = h(b′), γ(0) = g(e)

There is a mapϕ : E ′ −→ E(g, h)

ϕ(z) =(g′(z), hg′(z), h′(z)

)where hg′(z) denotes the constant path

The rectangle in (4.5) is called a Homotopy cartesian square, if the map ϕ is a homotopyequivalence.

In the framework of the Homotopy Category H omoTop, Homotopy cartesian is char-acterized as follows.

Proposition 4.1.27. Let

E ′ h′ //

g′

E

g

B′

h// B

be a commutative diagram of topological spaces.

This diagram a Homotopy cartesian square if and only if (E ′, g′, h′) is the Pullback of (g, h)

in the Homotopy Category H omoTop. (This means, the space E(g, h) described abovesatisfies the property of pullback, in H omoTop.)

Outline of the Proof. We need to prove, given a diagram

E ′h′ //

g′

E

g

B′

h// B

h′g ∼ g′h ∃ ϕ : E ′ −→ E(g, h) 3

E ′

g′

h′

&&

ϕ

##E(g, h) //

E

g

B′

h// B

commute in H omoTop and ϕ is unique in H omoTop.


Suppose H : E ′ × [0, 1] −→ B is a homotopy from gh′ to hg′. Define

ϕ(z) = (g′(z), H(z,−), h′(z)). Note H(z, 0) = gh′(z), H(z, 1) = hg′(z).

So, ϕ(z) ∈ E(g, h). The outer triangles commute (exactly). One needs to further prove, ifψ : E ′ −→ E(g, h) is another map, so that the outer triangles commute, up to homotopy,then ϕ ' ψ. The proof is complete.

Corollary 4.1.28. Let g : E −→ B be a continuous map and b ∈ B. Then, the followingcommutative diagram:

F (g, b) //

t=1

E

g

b // B

is a pullback diagram in H omoTop.

Definition 4.1.29. Suppose g : E −→ B be a continuous map and b ∈ B. Suppuse

F ι //

E

g

b // B

is a pullback diagram in H omoTop.

Then, we say that F ι // Eg //// B is a homotopy fibration.

1. From uniqueness of pullback, F ' F (g, b) are homotopicaly equivalent.

2. Therefore, for x0 ∈ F there is long exact sequence

· · · // πn (F, x0) // πn (E, ι(x0)) // πn (B, ι(b)) // πn−1 (F, x0) // · · ·

The following is an useful lemma.

Lemma 4.1.30. If B is contractible, and g : E −→ B is a map, then the map

π : F(g, b) = E ×B BI × b −→ E are homotopy equivalences.

Outline of the Proof. We construct the homotopy inverse as follows:

H : B′ × [0, 1] −→ B be 3 H(−, 1) = 1B, H(−, 0) = Cb

Let h : B −→ BI be defined by h(x) = H(x,−). Now define

ϕ : E −→ F(g, b) ϕ(z) = (z, hg(z), b) . So πϕ = 1E.


Also, need to prove

ϕπ(z, γ, b) = ϕ(z) = (z, hg(z), b) ∼ 1F(g,b) which we skip!

Back to Homotopy of Functors:

Now we try to adapt some of the above to categories and functors.

1. Let F : C −→ D be a functor. Let Y ∈ Obj(D). Let

j : Y \ F −→ C 3 j(X, v) = X where v : Y −→ FX

So, the commutative triangle

Y \ F j //

Fj ""

C

F

D

=⇒B (Y \ F )

B(j) //

B(Fj) %%

B (C )

B(F )

B (D)

LetCY : Y \ F −→ D be the constant functor Y.

This induces, the constant map, of the classifying spaces:

cY : B (Y \ F ) −→ B (D)

2. There is a natural transformation

CY −→ Fj by v : CY (X, v) = Y −→ Fj(X, v) = FX

So, by Proposition 4.1.9, there is a homotopy

H : B (Y \ F )× [0, 1] −→ B (D) ∗ = Y 7→ B (Fj)

So, we obtain a canonical map

B (Y \ F ) −→ F(B(Fj), Y ) sending z 7→ H(z,−) (4.6)

a path starting from Y to B(Fj)(z).

3. Is this map (4.6) homotopy equivalence? We answer this question subsequently.

4. Here is a definition borrowed from topology.


Definition 4.1.31. A commutative diagram, of categories

C

// D

E ′ // D′

is called Homotopy Cartesian, if

BC

// BD

BE ′ // BD′

is so.

In this case, if E ′ is contractible, we say C // // D // D′ is a homotopy fibration.Consequently, a long exact sequence of homotopy groups/set would follow.

Now we are ready to state Theorem B.

Theorem 4.1.32 (Theorem B). Let F : C −→ D be a functor such that

∀ Y −→ Y ′ ∈MorD(Y, Y ′) the induced functor Y ′ \ F −→ Y \ F

is a homotopy equivalence. Then, ∀ Y ∈ Obj(D), the commutative diagram

Y \ F j //

F ′

C

F

Y \ D

j′// D

j(X, v) = X

F ′(X, v) = (FX, v)

j′(Y ′, v) = Y ′is homotopy cartesian. (4.7)

The following would be consequences of the above:

1. Note, ∀ Y ∈ Obj(D), the category Y \ D has an initial object (Y, 1Y ). So, Y \ D iscontractible. Therefore, the upper right part of the cartesian square (4.7) would bea homotopy fibration.

2. Consequently, there ∀ Y ∈ Obj(D) and X ∈ Obj(C) with FX = Y

(so, X := (X, 1Y ) ∈ Obj(Y \ F )), there is a long exact sequence:

· · · // πi+1 (D, Y ) // πi

(Y \ F, X

)// πi (C, X) // πi (D, Y ) // · · ·

3. As always, this theorem admits a dual formulation, replacing the categories Y \F bycategories F/Y etc., in the diagram (4.7).

A Corollary to Theorem B, inn analogy to Corollary 4.1.24:


Corollary 4.1.33. Suppose F : C −→ D is a prefibred functor. Assume, for every arrowu : Y −→ Y ′ in D , the base change functor u∗ : F−1(Y ) −→ F−1(Y ′) is a homotopyequivalence. Then, ∀ Y ∈ Obj(D) the diagram

F−1(Y ) //

C

pt // D

is homotopy cartesian. (We say that the upper right sequence is a homotopy fiber.)

Consequently, for any X ∈ Obj(F−1(Y )), there is a long exact sequence:

· · · // πi+1 (D, Y ) // πi (F−1(Y ), X) // πi (C, X) // πi (D, Y ) // · · ·

Further, same holds, if we replace the word "prefibred", by "precofibred".

Proof. To apply Theorem B (4.1.32), we need to establish that Y ′ \ F −→ Y \ F is ahomotopy equivalence. We have the diagram of prefibred spaces:

F−1(Y ′) //

%%

Y \ F

F−1(Y )

The vertical functor is a homotopy equivalence, by the definition of prefibred spaces. Thediagonal map is a homotopy equivalence, by hypothesis of the corollary. Therefore, thehorizontal functor is a homotopy equivalence. Now, we have another commutative diagram:

F−1(Y ′) //

Y \ F

Y ′ \ F

99

The vertical map is also a homotopy equivalence, by taking u = 1Y ′ (by prefibred property).So, the diagonal map is homotopy equivalence. Now, the corollary follows by an applicationof Theorem B (4.1.32).

We proceed to give proof of Theorem B (4.1.32):First, we have the following definitions.

Definition 4.1.34. Let f : E −→ B be a map of topological spaces. Such a map is calleda quasi-fibration, if

∀ b ∈ B, the natural map f−1(b) −→ F(f, b) is a weak equivalence (see Defn.2.5.11).


Recall the natural map ι, given by the diagram

f−1(b)

ι

$$ &&

!!

F(f, b) //

E

f

b // B

It follows from Whitehead’s Theorem 2.5.12, if E, B, f−1(b) are CW complexes, g isquasi-fibration if and only if f−1(b) −→ F(f, b) is a homotopy equivalence. Hence, in suchcases

f−1(b) //

E

f

b // B

is homotopy cartesian.

For our purpose, this would be the main point of introducing quasi-fibrations.


Lemma 4.1.35. Suppose I is a small category and I −→ Top is a functor, with i 7→ Xi.

1. Let N I be the nerve of I

2. Consider the simplicial topological space

p 7→ Xp :=∐

σ∈N Ip

Xσ(0). Let XI := |Xp| be its geometric realization.

We have a map of simplicial spaces γ : Xp −→ N I, given by the associations:

p //

!!

Xpγp

N Ip

3. The map γ of simplicial sets, gives a map g := |γ| : XI −→ BI, of topological spaces.

Assume, ∀ i −→ j ∈ MorI , the induced arrow Xi −→ Xj is a homotopy equivalence.Then, g is a quasi-fibration.

Proof. The proof is given by proving the same to the restriction of g, to g−1Fp −→ Fp,where Fp is the p-skeletons of BI (see Section 3.3.1). That would suffice, by Lemma 1.5 of[Dold-Lashof]. We have a commutative diagram of push out (co-cartesian) diagrams:∐

σ∈NIp′σ × |∂∆[p]| ×Xσ(0)

// _

**

g−1Fp−1 _

g

##∐σ∈NIp′

σ × |∂∆[p]| // _

Fp−1 _

∐σ∈NIp′

σ × |∆[p]| ×Xσ(0)//

**

g−1Fpg

##∐σ∈NIp′

σ × |∆[p]| // Fp

Here, the front of the cube is the pushout diagram defining Fp and backside is the pushoutfor the p-skeleton Xp

I . The rest of the proof given in [Q] consists of multiple cross referencesto the paper of [Dold-Lashof]. We would avoid getting in to deeper details of the same.


Proof to Theorem B (4.1.32): We recall some of the notations from Theorem A (4.1.22).Let S(F ) be the category, described below:

1. The objects are

Obj(S(F )) = (X, Y, v) : X ∈ Obj(C ), Y ∈ Obj(D), v ∈MorD (Y, FX)

2. For (X, Y, v), (X ′, Y ′, v′) ∈ Obj(S(F )), Define MorS(F ) ((X, Y, v), (X ′, Y ′, v′)) =(u,w) ∈MorC (X,X ′)×MorD(Y ′, Y ) :

Y ′ w //

v′

Y

v

FX ′ FXFuoo

commute

There are functors:

S(F )p1 //

p2

C

Do

3p1(X, Y, v) = X p1(u,w) = up2(X, Y, v) = Y p2(u,w) = w

As in the proof of Theorem A (4.1.22), p1 is a homotopy equivalence.

Also, from the proof of Theorem A (4.1.22), the map B(p2) : BS(F ) −→ B(Do) is therealization of the map∐Yp // Yp−1

// · · · // Y0

B(Y0 \ F ) −→∐

Yp // Yp−1// · · · // Y0

pt = N (D0)p

By hypothesis (4.1.32), Lemma 4.1.35 is applicable, with I := Do, to the functor Y 7→B(Y \ F ) from Do −→ Top. By Lemma 4.1.35, we see that Bp2 is a quasi fibration. Sinceall the spaces concerned are CW complexes, by Whitehead’s Theorem 2.5.12:

∀ Y ∈ Obj(Do)

Y \ F //

S(F )

p2

pt

Y// Do

is a homotopy cartesian.

Now, we have the following diagram:

Y \ F //

S(F ) ∼ //

F ′

C

F

Y \D //

o

S(1D) ∼//

o

D

ptY

// Do

All the three rectangles are actually cartesian. The combination of two rectangles on theleft is a homotopy cartesian. So, the upper left rectangle is homotopy cartesian. Also,two horizontal maps on the right hand rectangle are homotopy equivalence. Thereforecombined two horizontal rectangles is homotopy cartesian.


4.2 The K-Groups of Exact Categories

This is an exposition of [Q, Section 2] form Quilen’s paper. Given an exact category E ,Quillen [Q], associates a new category QE , as follows.

4.2.1 Quillen’s Q-Construction

Definition 4.2.1. Suppose E is a small exact category. Define a new category QE asfollows.

1. The objects of QE are same as that of E .

2. Give two objects X, Y ∈ QE , a morphism X −→ Y in QE is defined as follows:Consider diagrams, as follows:

X Zpoooo i // Y 3 ∃ conflations

K ′ // Z

p // // X

Z

i// Y // // C

in E . (4.8)

In other words, p is a deflation and i is an inflation. It is sometimes convenient todisplay this diagram as:

Z

p

i// Y

X

Two such diagrams (as in (4.8)),X Z

poooo i // Y

X Z ′p′oooo

i′// Y

are said to be isomorphic, if

∃ isomorphism τ : Z∼−→ Z ′ 3

Z pi

p

~~~~o τ

X Y

Z ′. i′

>>

p′

````commutes.

A morphism X −→ Y in QE is an isomorphism class of diagrams, as in (4.8). Adiagram, as in (4.8), will be denoted by (Z, p, i).

4.2. THE K-GROUPS OF EXACT CATEGORIES 111

3. (Compositions): Let X −→ Y and Y −→ Z be two morphisms in QE , repre-sented by X W

poooo i // Y , Y Vqoooo j // Z , the composition is represented by

(V, pq′, ji′), as in the diagram:

U i′ //

q′

V

q

j // Z

W

i//

p

Y

X

where (U, i′, q′) is the pullback. (4.9)

We remark, composition is well defined and associative. It is also imperative that weassume that the isomorphism classes of such diagrams (4.9), form a set.

We list the following:

1. Recall that an inflation A → X in E is called an admissible monomorphism and adeflation X B in E is called an admissible epimorphism.

2. Given an object X in E , by a sub object K of X, we mean an isomorphism classesof admissible monomorphism K → X, as described in the following commutativediagram

K //

o

X

σo

K ′ // X

, which is equivalent to

K //

o

X

K ′ // X

Likewise, by an admissible quotient C of X, we mean an isomorphism classes ofadmissible epimorphisms X C, as described in the following commutative diagram

X

σo

// // C

o

X // // C ′

, which is equivalent to

X // // C

o

X // // C ′

There is a bijection between sub objects of X and quotient objects of X.

3. The set of all sub objects [K] of X forms a partially ordered set. We say [K ′] ≤ [K](or K ′ ≤ K) if there is an admissible monomorphism ι (unique):

K ′ _

∃ ι

// X

K // X

If K ′ ≤ K, we say (K ′, K) is an admissible layer of sub objects of X. In this caseK/K ′ is admissible.


4. It follows, a morphism X −→ Y is QE is a pair ((K,Z), θ) of admissible layers of Y ,and an isomrphism θ : X : X

∼−→ Z/K. Diagramtically:

K _

Z ι //

p

Y

X

where K = ker(p).

5. To see how composition works with respect to this description of morphisms in QE ,via admissiible layers, refer to the diagram (4.9):

(a) The map X −→ Y is represented by(

(ker(p),W ), Wker(p)

∼−→ X),

(b) the map Y −→ Z is represented by(

(ker(q), V ), Vker(q)

∼−→ Y).

(c) The composition X −→ Z is represented by(

(ker(pq′), U), Uker(pq′)

∼−→ X)

Diagrammatically, extending (4.9):

ker(q) _

lL

yy

ker(q) _

ker(pq′)

// U i′ //

q′

V

q

j // Z

W

i//

p

Y

X

Write

KV := ker(q)KU := ker(pq′)

(4.10)

6. in QE , morphisms 0 −→ Y are is one to one correspondence with admissible subobjects U → Y . In terms of layers they are ((U,U), ∗).

7. Give an admissible monomorphism ι : X → Y in E , the corresponding map in QEis denoted by ι! : X −→ Y . In [Q], such a morphism in QE was referred to as"injective" (which is not to be confused with "injective maps" in the usual sense).

8. Likewise, given an admissible epimorphism ζ : Y X in E , it gives rise to amorphism ζ ! : X −→ Y in QE . In [Q], such a morphism in QE was referred to as"surjective" (which is not to be confused with "surjective maps" in the usual sense).

9. By definition, any morphism u : X −→ Y in QE can be written as u = ι!p!. This

factorization is unique up to isomorphism (by definition):

X U

o

poooo ι // // Y

X Vqoooo

ζ// // Y


10. Given a bicartesian square (i.e. pullback and pushout)

U ι //

ζ

Y

η

X

β// Z

=⇒ u = ι!ζ! = η!β! (4.11)

This follows directly from the definition of η!β!. The latter factorization is also unique(which follows from the definitions of compositions and equality of maps.) .

11.

Lemma 4.2.2. If a map is both injective and surjective, then it is an isomorphism.Proof. Suppose u = Θ! = Ψ!. Then, we have the commutative diagram:

X X1Xoooo Θ //

Φo

Y

X Y

1Y//

Ψoooo Y

This shows that Θ = Ψ−1. Now, we prove that if Θ is an isomorphism in E , the Θ!

is an isomorphism in QE and its inverse is Θ−1)!:

X X1Xoooo Θ //

Θ

Y

X Y

1Y//

Θ−1oooo Y


12.

Lemma 4.2.3. Every morphism in QE is a monomorphism.Proof. Since every map in QE is composition of an "injective" followed by a "sur-jective" map, prove they are mono in QE . Suppose η : Y → W is a monomorphismand β : Y W is a epimorphism in QE .Consider two maps in QE :

u : X Zpoooo ι // Y

v : X Z ′qoooo ζ // Y

Now suppose η!u = η!v. Then, we have the combative diagram:

η!u : X Z

ϕo

poooo ι // Y η //W

η!v : X Z ′qoooo ζ // Y

η //W


where ϕ is an isomorphism. Since, η is a mono, ζϕ = ι and hence u = v. Likewise,we check that β!u = β!v implies u = v. The proof is complete.

13. For an object Z in E , the categoryQE \Z is equivalent to the ordered set of admissiblelayers (KU , U) of Z. It follows from Equation 4.10 that the ordering is given by

(KU , U) ≤ (KV , V ) if KV ≤ KU ≤ U ≤ V

Diagramatically,

14. The Universal Property of QE :First, we have the following observations:

(a) There is No functor E −→ QE because, given a morphism f : X −→ Y in E ,there is no natural map in QE associated to f .

(b) However, given a admissible ι : X → Y monomorphism, in E , we can associateι! in QE . Given two composable admissible ι, ζ monomorphism, in E , we have(ζι)! = ζ!ι! in QE .Likewise, given a admissible p : X Y epimorphism, in E , we can associatep! in QE . Given two composable admissible p, q epimorphism, in E , we have(pq)! = q!p! in QE .Further (IdX)! = (IdX)! = IdX .

(c) Given the bicartesian square (4.11), where all the maps are admissible (epi ormono), then ι!ζ ! = η!β!.

Now, let C be a category. Consider the diagram of associations (not functors):

E

h !!

// QE

C

3

X 7→ hX ∀ X ∈ Obj(E )ι 7→ ι! : hX −→ hY ∀ admissible mono ι : X → Yp 7→ p! : hY −→ hX ∀ admissible epi p : X Y

While h is a not a functor, the above defines two functors:

(a) E− −→ C where E− is the category whose objects are same as those of E , andmorphisms are the inflations of E ,

(b) E+ −→ C where E+ is the category whose objects are same as those of E , andmorphisms are the deflations of E .

Further assume that conditions 14b, 14c hold. Then, there is a unique functor Θ, asfollows:

E

h !!

// QE

Θ

C

”commute” and compatible with

ι 7→ ι!p 7→ p!

Finally, we have the obvious lemmas:


Lemma 4.2.4. For any exact category E , we have

QE = Q(E o)

induced by the contra variant functor E −→ E o.

4.2.2 The Higher K-groups of an Exact Category

First, we define the Grothendieck group of an exact category.

Definition 4.2.5. Suppose E is a small exact category. Let Z(E ) be the free abelian groupgenerated by the objects of E . Let R(E ) be the subgroup of Z(E ) generated by the set

Y −X − Z ∈ Z(E ) : X // Y // // Z is a conflation.

Define the Grothendieck group

K0(E ) :=Z(E )

R(E )

We Remark:

1. The conflation 0 // 0 // // 0 implies 0 = [0] ∈ K0(E ).

2. Suppose π : X∼−→ Y is an isomorphism. Then, 0

// Xπ // // Y is a conflation.

Hence [X] = [Y ] ∈ K0(E ).

3. The map ι : E −→ K0(E ) has the following universal property:Suppose G is an abelian group and β : E −→ G is a set theoretic map, such that

ζ(Y ) = ζ(X)ζ(Z) ∀ conflations X // Y // // Z

Then, there is a group homomorphism

ψ : K0(E ) −→ G 3Obj(E ) ι //

β%%

K0(E )

ψG

commutes.

We say that K0(E ) has the above universal property, in the category Ab of abeliangroups. We will see next, that K0(E ) has the same universal property, in the categoryGr of groups.


4. The above definitionK0(E ) makes sense, whenever E has a set of isomorphism classesof objects, by replacing E by an equivalent small exact subcategory E ′ ⊆ E .

In this case (and otherwise), let E be the set of all isomorphism classes of objectsin E and Z(E) be the free abelian groups generated by E. Let R(E) ⊂ Z(E) bedefined as above. Then,

K0(E ) =Z(E)

R(E).

Lemma 4.2.6. Let E be a small exact category. Then, ι : Obj(E ) −→ K0(E ) has thefollowing universal property:

Given any (possibly, non-commutative) group G and a set theoretic map ζ : Obj(E ) −→G such that

ζ(Y ) = ζ(X)ζ(Z) ∀ conflations X // Y // // Z

there is a group homomorphism

ψ : K0(E ) −→ G 3Obj(E ) ι //

ζ%%

K0(E )

ψG

commutes.

In particular,

K0(E ) :=Z(E )

R(E )∼=

F (E )

N (E )

where F (E ) is a the free (non-abelian group generated by Obj(E ) and N (E ) is the normalsubgroup generated by the set

W =Y (XZ)−1 : X

// Y // // X is a conflation.

Proof. Write G (E ) := F (E )N (E )

. (Recall, elements of F (E ) are "words".) Note that the mapη : Obj(E ) −→ G (E ) has the universal property in the category Gr of groups. Using thesplit exact conflations, it follows η(X)η(Y ) = η(X ⊕ Y ) = η(Y ⊕X) = η(Y )η(X). Formthis, it follows, G (E ) is abelian. Therefore, both G (E ) and K0(E ) have same universalproperty in the category Ab of abelian groups. Hence G (E ) ≡ K0(E ).

The following proposition establishes the key connection between classical K-theory tohomotopy groups of BQE .

Proposition 4.2.7. Let E be a small Exact category. The classifying space BQE isconsidered as a pointed topological space, the base point being the 0-object of E . Then,


1. The classifying space BQE is (path) connected. So, π0 (BQE ) = 0.

2. There is an isomorphism K0(E )∼−→ π1 (BQE , 0).

Proof. The following proof is drawn from [Sm1], which is more elaborate than the one in[Q]. Suppose X is an object of E or QE . The diagram

0 0oooo

0X// X represents a map 0 −→ X in QE .

This gives a path from 0 to X. Since all the vertices of BQE are path connected to 0,QE is path connected. Hence π0 (BQE ) = 0. (We comment that MorQ(E )(0, X) is notnecessarily a singleton.)

With notations, as Lemma 4.2.6,

K0(E ) ≡ F (E )

N (E )

Define a group homomorphism ϕ0 : F (E ) −→ π1 (BQE ), as follows. For an arrow → inQE , the corresponding path in the classifying space B(QE ) will be denoted by γ(→).

Suppose X ∈ Obj(E ). Consider two morphisms 0 −→ X in QE :

(0, 0, 0X) : 0 0oooo 0X // X

(X, 0, 1X) : 0 Xoooo 1X // X

By Lemma 4.1.3, these two morphisms defines two paths γ0 := γ((0, 0, 0X), γ1 := γ((X, 0, 1X)from 0 to X in BQE . So, `X := γ−1

0 γ1 is a loop. Define

ϕ0 : F (E ) −→ π1 (BQE ) by ϕ(X) = [`X ]

Now, consider a conflation

X i // Y

p // // Z which corresponds to Y (XZ)−1 ∈ N (E ).

Now, `X = γ(0, 0, 0X)−1γ(X, 0, 1X) : 0 ++33 X is homotopic to

γ ((X, 1X , i)(0, 0, 0X))−1 γ ((X, 1X , i)(X, 0, 1X)) : 0γ(0,0,0X)

++

γ(X,0,1X)

33 Xγ(X,1X ,i) // Y

The compositions (X, 1X , i)(0, 0, 0X) is given by

0 0X /

0

X

1X

i // Y

0

0X/

0

X

0

which is (0, 0, 0Y )


Likewise, the compositions (X, 1X , i)(0, 0, 1X) is given by

X 1X /

1X

X

1X

i // Y

X

1X/

0

X

0

which is (X, 0, i)

So, [`X ] = [γ(0, 0, 0Y )−1γ(X, 0, i)]. Likewise, [`Z ] = [γ(X, 0, i)−1γ(Y, 0, 1Y )]. So, [`X ][`Z ] =[`Y ]. Therefore, ϕ0 factors through a well defined homomorphism ϕ : K0(E ) −→ π1 (BQE , 0).

Now, we prove that ϕ is surjective. Let ` ∈ π1 (BQE , 0). By Proposition 2.5.13, the mapπ1 (BQE 1, 0) π1 (BQE , 0) is surjective, where BQE 1 denotes the one skeleton. Therefore` is represented by a product of paths γnγn−1 · · · γ1, such that

1. γi is a path from Xi−1 to Xi, with Xi ∈ Obj(E ), and X0 = Xn = 0.

2. Further,

(a) either γi = γ(fi), in which case fi ∈MorQE (Xi−1, Xi),

(b) Or, γi = γ(fi)−1, in which case fi ∈MorQE (Xi, Xi−1).

3. In fact, for any nonzero object X in E , MorQE (X, 0) = φ. So, we can assumef1 : 0 −→ X1, and fn : 0 −→ Xn−1.

Let gi = γ(Xi, 0, 1Xi). Up to homotopy, we have

γnγn−1 · · · γi · · · γ1 = (γngn−1) · · ·(g−1i γigi−1

)· · ·(g−1

2 γ2g1

) (g−1

1 γ1

)We will prove that, each

[(g−1i γigi−1

)]is in the image of ϕ. There are two cases,

1. γi = γ(fi), where fi : Xi−1 −→ Xi. So, fi is given by

Xi−1 Uipoooo

ι// Xi

We have, gi−1 = γ(Xi−1, 0, 1Xi−1) is given by the arrow

0 Xi−1oooo Xi−1

So, γigi−1 is given by the arrow

0 Uioooo

ι// Xi

Therefore,

g−1i γigi−1 = γ(Xi, 0, 1Xi)

−1γ(Ui, 0, ι) ∼ γ(Xi, 0, 1Xi)−1γ ((Ui, 1Ui , ι)o(Ui, 0, 1Ui))


∼ γ(Xi, 0, 1Xi)−1γ(Ui, 1Ui , ι)γ(Ui, 0, 1Ui)

∼ γ(Xi, 0, 1Xi)−1γ(Ui, 1Ui , ι)γ(0, 0, 0Ui)γ(0, 0, 0Ui)

−1γ(Ui, 0, 1Ui)

∼ γ(Xi, 0, 1Xi)−1γ ((Ui, 1Ui , ι)o(0, 0, 0Ui)) γ(Ùi)

∼ γ(Xi, 0, 1Xi)−1γ(Xi, 0, 0Xi)γ(Ùi) ∼ γ(`Xi)

−1γ(Ùi)

So, [(g−1i γigi−1

)]= ϕ(Xi)

−1ϕ(Ui)

2. γi = γ(fi)−1, where fi : Xi −→ Xi−1. This case is dealt similarly, as above.

This establishes that ϕ is surjective.

To prove that ϕ is an injective, we define a map π1 (BQE , 0) −→ K0(E ). As in Example4.1.5, let K0(E ) denote the category with one object ? and Mor(?, ?) = K0(E ). Define afunctor

Φ : QE −→ K0(E ) ∀ X ∈ Obj(QE ) Φ(X) = ? and,

for f :=

W

p

ι // Y

X

∈MorQE (X, Y ) Φ(f) = [ker(p)].

We check that Φ is a functor. For another morphism

g :=

V

q

ι // Z

Y

∈MorQE (Y, Z) composition gf is displayed in Diagram 4.10

We need to prove that [ker(p)] + [ker(q)] = [ker(pq′)]. This follows, because

ker(q) // ker(qp′) // // ker(p) is a conflation.

Therefore, Φ is a functor. Hence, it induces a map

η := π1 (BΦ) : π1 (BQE , 0) −→ π1

(K0(E ), ?

)= K0(E )

Now, for [X] ∈ K0(E ),

η(ϕ([X])) = η(γ(0, 0, X)−1γ(X, 0, 1X)) = −[0] + [X] = [X].

Therefore, ϕ is a bijection. The proof is complete.


4.2.3 Higher K-Groups

Definition 4.2.8. Suppose E is a small exact category and 0 denote the/a zero object.

∀ i ∈ N define Ki(E ) = πi+1 (BQE , 0)

We have the following remarks:

1. The definition is independent of the choice of the zero object, because BQE is pathconnected.

2. The definition extend to any exact category E that has a set of isomorphisms classesof objects.

Lemma 4.2.9. Any exact functor F : E −→ D , of exact categories induces a functorQE −→ QD . This induces homomorphisms of K groups

Fi := Ki(F ) : Ki(E ) −→ Ki(D) ∀ i ≥ 0

1. So, for each (fixed) i ≥ 0, the association

i 7→ Ki(E )

is a functor, from the "category" of exact categories and exact functors to the categoryof abelian groups.

2. If F : C∼−→ D is an equivalence of categories, then Fi : Ki(E )

∼−→ Ki(D) areisomorphisms, ∀ i ≥ 0.

3. For any exact category E , by Lemma 4.2.4, we have

∀ i ≥ 0 Ki(Eo) = πi+1(BQ(E o)) = πi+1(B(QE )) = Ki(E ).

4. Suppose E ,D are two exact categories. Then, E × D is also an exact category. Itfollows Q(E ×D) = Q(E )×Q(D).Further, it follows B(Q(E ×D)) = B(Q(E ))× B(Q(D)). Therefore,

∀ i ≥ 0 Ki(E ×D) ' Ki(E )⊕Ki(D).

5. Suppose E is an exact category. Then the direct sum functor E × E −→ E , sending(M,N) 7→M ⊕N is an exact functor. This induces a group homomorphism

∀ i ≥ 0 ⊕∗ : Ki(E )⊕Ki(E ) −→ Ki(E )


The map coincides with the usual direct sum map because of the commutative dia-gram:

Ki(E )

i1

1

##Ki(E )⊕Ki(E ) ⊕∗

// Ki(E )

Ki(E )

i2

OO

1

;;

where i1 is induced by the functor M 7→ (M, 0) and i2 is induced by the functorM 7→ (0,M).

6. Suppose I is a small filtering category and j 7→ Ej functor from I to the category ofsmall exact categories.

Let lim→ Ej be the inductive limit ([Q, Prop 3, pp 84]), which is an exact category.Then, Q(lim→ Ej) = limj (Q(Ej)). Hence,

∀ i ≥ 0 Ki

(lim→

Ej)

= lim→Ki(Ej).

Example 4.2.10. Let A be a ring with unity and P(A) be category of all projective (left)A-modules. We write,

∀ i ≥ 0 Ki(A) := Ki (P(A))

1. Given a ring homomorphism A −→ B induces a functor P(A) −→ P(B), sendingP 7→ B ⊗ P . This induces well defined group homomorphism

∀ i ≥ 0 Ki (A) −→ Ki(B).

2. This makes Ki(A) a covariant functor from the category of rings with unit, to thecategory of abelian groups.

3. It follows,∀ i ≥ 0 Ki(A×B) = Ki(A)⊕Ki(B)

4. Suppose I is a small filtering category and j 7→ Aj functor from I the category ofrings with unit. It follows, from (6):

∀ i ≥ 0 Ki

(lim→Aj

)= lim→Ki(Aj).

Example 4.2.11. Let A be a noetherian commutative ring with unit.


1. The category QCoh(A) be the category of A-modueles, is an exact category. So, thisgives rise to a K-theory.

2. The category Coh(A) be the category of finitely generated A-modules, is an exactcategory. So, this gives rise to a K-theory. This is usually referred to as G-theory ofA and the K-groups are denoted by Gi(A).

3. Then, we have Example 4.2.10. The category P(A) be the category of finitelygenerated projective A-modules over X, is an exact category. So, this gives rise toa K-theory. This is usually referred to as K-theory of X and the the K-groups aredenoted by Ki(A).

Comparison of the K-Theory of these categories constitutes a big chunk of the literature.

Example 4.2.12. More generally, let X be (noetherian) scheme.

1. The category QCoh(X) be the category of quasi coherent modules over X, is anexact category. So, this gives rise to a K-theory.

2. The category Coh(X) be the category of quasi coherent modules over X, is an exactcategory. So, this gives rise to a K-theory. This is usually referred to as G-theory ofX and the the K-groups are denoted by Gi(X).

3. The category P(X) be the category of locally free modules over X, is an exactcategory. So, this gives rise to a K-theory. This is usually referred to as K-theory ofX and the K-groups are denoted by Ki(X).

Comparison of the K-Theory of these categories constitutes a big chunk of the literature.

Lemma 4.2.13. Let A be a commutative noetherian ring. The classical definition ofK1(A), which we denote, temporarily, by Kc

1(A), is as follows:

Kc1(A) =

GL∞(A)

EL∞(A)

Readers are referred to [GM] for further details. We have,

Kc1(A) =

GL∞(A)

EL∞(A)' K1(P(A))

Proof. Skip. See [Q, pp. 96], for the statement.

4.3. CHARACTERISTIC EXACT SEQUENCES AND FILTRATIONS 123

Lemma 4.2.14. Let A be a commutative noetherian ring. A classical definition of K2(A),which we denote, temporarily, by Kc

2(A), is obtained through the so called "SteinbergGroups", given in [Mi]. We have,

Kc2(A) ' K2(P(A))

Proof. Skip. See [Q, pp. 96], for the statement.

4.3 Characteristic Exact Sequences and Filtrations

Let E be an exact category and let ε(E ) be the category of the exact sequences in E .More precisely, Objects X, Y of ε(E ) are the exact sequences (i.e. conflations) in E , as thediagram and a morphism f : X −→ Y is a commutative diagram in E :

X :

f

0 // K

ι

//M

β

// // C

ϕ

// 0

Y : 0 // K ′ //M ′ // // C ′ // 0

(4.12)

1. Give X ∈ Obj(ε(E )) as in (4.12), write s(X) := K, t(X) = M and q(X) = C. Then,s, t, q : ε(E ) −→ E defines three functors. s(X), t(X), q(X) would be called "sub","total" and "quotient" objects of X.

2. In fact, ε(E ) has a structure of an exact category, where a sequence X // Y // Zis declared exact, if the corresponding vertical sequences (as in diagram (4.12)) areexact.

3. It follows, s, t, q : ε(E ) −→ E are exact functors.

Theorem 4.3.1. Let E be an exact category. Then, the functor (s, q) : Q(ε(E )) −→Q(E )×Q(E ) is a homotopy equivalence.

Proof. By Theorem 4.1.22, it would be enough to show that (s, q)/(K,C) is contractible,for any pair of objects (K,C) of Q(E ).

Fix such a pair of objects (K,C) and let C := (s, q)/(K,C). So,

C := (s, q)/(K,C) = (X, u, v) : u ∈MorQE (sX,K), v ∈MorQE (qX,C)

More explicitly, u : sX U

poooo i // K

v : qX Vqoooo j // C

(4.13)


1. Let E ′ be the full subcategory of C , consisting of objects (X, u, v) such that u issurjective, meaning u looks like

u : sX Koooo K

2. Let E ” be the full subcategory of C , consisting of objects (X, u, v) such that u issurjective and v is injective, meaning u, v looks like

u : sX Koooo K

v : qX qX // C

Lemma 4.3.2. The inclusion functors C ” −→ C ′ and C ′ −→ C have left adjoints.Consequently, C −→ C ” is a homotopy equivalence (by (4.1.10)).

Proof. First, we look at the functor C ′ −→ C . Given X = (X, u, v) in C , we will establishthat there is an universal arrow ι : X −→ X , with X ∈ C ′. This means, given Y ∈ C ′ andζ : X −→ Y in C , as in the diagram, there is a unique morphism η, as in the commutativediagram:

X

ζ

ι // X∃ uniqueη

YAs in Equation 4.13,

u : sX U i //poooo K =⇒

sX U i //

p

poooo K // //

j′

W

sX

i′//M // //W

=⇒ u = (j′)!(i′)!

where W is the co-kernel of i andM is the pushout. Then, u = j!i!. Define X , as follows:

X sX _

i′

// tX _

// // qX

X := M // T // // qX

where T is the pushout. Therefore s(X) =M, t(X) = T , q(X ) = qX. So, j′ : K s(X),induce the "surjective" map (j′)! : s(X) −→ K in QE . Further, v : q(X) −→ C. DefineX := (X, j!, v). Then, X is in C ′. One checks, X −→ X is universal.

It follows, from universality, for X in C and Y in C ′,

MorC (X ,Y)∼−→MorC ′(X ,Y)

This means, X 7→ X is a Left Adjoint of the inclusion C ′ −→ C .


Now, we prove that C ” −→ C ′ has a left adjoint. Let X = (X, u, v) be an object in C ′,where u : sX −→ K is surjective and v : qX −→ C is a morphism in QE , which is givenby

v : qX Ujoooo i // C

We essentially dualize the above argument. Consider the diagram:

X : 0 // sX // T //

U //

j

0

X : 0 // sX // tX // qX // 0

where T is the pull back. Write X = (X, u, i!). One checks that X −→ X is universal.This again establishes that C ” −→ C ′ has a left adjoint, namely , X 7→ X .


Proof of Theorem 4.3.1: By Lemma 4.3.2, it would suffice to show that C ” has an initialobject. Let jK : K 0 and iC : 0 −→ C be the unique maps. Let X0 = (0, j!

K , (iC)!). Onechecks that X0 is an initial object in C ”.


4.3.1 Additivity Theorem

Theorem 4.3.3. Let E ,D be two exact categories. Let G,F,H : E −→ D be three exactfunctors, such that

0 // G // F // H // 0

be exact. Then,

∀ i ≥ 0 F∗ = G∗ +H∗ : Ki(E ) −→ Ki(D)

Proof. As before, let ε(D) denote the exact category of short exact sequences in D . Lets, t, q : ε(D) −→ D be the sub, total and quotient functors. Then,

0 // s // t // q // 0 is exact.

First, we prove that t∗ = s∗ + q∗ : Ki(ε(D)) −→ Ki(D). Let f : D × D −→ ε(D) be thefunctor sending (K,C) to the split exact sequence:

0 // K // K ⊕ C // C // 0

Note (s, q)f = Id. By Theorem 4.3.1, (s, q) is a homotopy equivalence Q(ε(D)) −→Q(D)×Q(D). Hence, f is also a homotopy equivalence Q(D)×Q(D) −→ Q(ε(D)).


We have

tf = ⊕(s, q)f :

QD ×QDf //

f

Qε(D)

t

Qε(D)

(s,q)

QD ×QD ⊕

// QD

So,t∗f∗ = ⊕∗(s∗, q∗)f∗ = (s∗ + q∗)f∗

Since f∗ is isomorphism, we have t∗ = s∗ + t∗.

To prove the theorem, consider the exact functor

ϕ : E −→ ε(D) ending X 7→(

0 // G(X) // F (X) // H(X) // 0)

Now, for integers i ≥ 0, we have consider the maps Ki (E ) −→ Ki (D). We have

F∗(X) = (tϕ)∗(X) = t∗(ϕ∗(X)) = s∗(ϕ∗(X))+q∗(ϕ∗(X)) = (sϕ)∗(X)+(qϕ)∗(X) = G∗(X)+H∗(X).


Corollary 4.3.4. Let Φi : C −→ D be exact functors, for i = 0, 1, . . . , n. Suppose

0 // Φ0// Φ1

// Φ2// · · · Φn

// 0

is an exact sequence. Then,

n∑p=0

(−1)p (Φp)∗ = 0 : Ki(E ) −→ Ki(D)

Proof. Follows from Theorem 4.3.3, by inducetion.

Filtrations of Functors

Let E ,D be two exact categories (small).

1. Let C = Fun(E ,D) be the category al all exact functors, whose objects are exactfunctors and a morphism Φ : F −→ G is a natural transformation.


2. In fact, C = Fun(E ,D) is an exact category, where a sequence

0 // Φ // Ψ // Θ // 0

is declared short exact, if

∀ X ∈ Obj(E ) 0 // Φ(X) // Ψ(X) // Θ(X) // 0 is exact.

3. A sequence of natural transformations (morphisms), in C ,

0 = Φ0 // Φ1

// Φ2 // · · · Φn = Φ

is defined to be, an admissible filtration, if

∀X ∈ Obj(E ), ∀ i = 0, . . . , n−1 Φi(X) // Φi+1(X) is an admissible monomorphism.

4. Given an admissible filtration, as above, the quotient morphisms Φi+1

Φiare defined,

determined up to canonical isomorphism. There does not seem to be any reason whysuch a quotient would be exact. However, if Φi+1

Φiis exact ∀ i = 0, 1, . . . , n− 1, then

ΦpΦq

is exact, for all 0 ≤ q ≤ p ≤ n.

Corollary 4.3.5. Let Φ : C −→ D be an exact functor. Suppose Φ admits an admissiblefiltration

0 = Φ0 // Φ1

// Φ2 // · · · Φn = Φ

such that Φp+1

Φpis exact ∀ p = 0, 1, . . . , n− 1. Then,

Φ∗ =n∑p=1

(Φp

Φp−1

)∗

: Ki(E ) −→ Ki(D)

Proof. Follows from Theorem 4.3.3.

4.3.2 The Prime Examples

Example 4.3.6. Let A be commutative noetherian ring and P(A) be the category finitelygenerated projective A-modules. Then, we write Ki(A) := Ki(P(A)). This example wasdiscussed in (4.2.10).

1. Given a projective A-modules P , the Q 7→ P ⊗ Q defines an exact functor P ⊗ :

P(A) −→P(A). This will induce homomorphisms:

[P ⊗−] : Ki(A) −→ Ki(A)


2. In fact, short exact sequences

0 // P−1// P // P1

// 0

induce exact sequences of exact functors:

0 // P−1 ⊗− // P ⊗− // P1 ⊗− // 0

So, it follows, from (4.3.3),

[P ⊗−]∗ = [P−1 ⊗−]∗ + [P1 ⊗−] : Ki(A) −→ Ki(A) ∀ i ≥ 0

Let X be a ringed space. Let P(X) be the category of locally free sheaves (i.e. categoryof locally free sheaves of OX-modules). Then, all of the above works for P(X).

Before we discuss the next example, we give the following graded versions of Nakayama’sLemma.

Lemma 4.3.7. Let A = A0 ⊕ A1 ⊕ A2 ⊕ · · · be a graded commutative ring and M =⊕n∈ZMn be a finitely generated graded A-module. Let A+ =

⊕i≥1Ai and M = A+M .

Then, M = 0.

Proof. Let m1,m2, . . . ,mk be a set of homogeneous generators of M . We havem1

m2

· · ·mk

=

a11 a12 · · · a1k

a21 a22 · · · a2k

· · · · · · · · · · · ·ak1 ak2 · · · akk

m1

m2

· · ·mk

where aij ∈ A are homogeneous elements of positive degree. Let A = (aij) be the k × kmatrix, as above. So,

(I − A)

m1

m2

· · ·mk

=

00· · ·0

Multiplying by the Adj(I − A), we have det(I − A)mi = 0 ∀ i = 1, . . . , k. It follows,det(A) = 1 + f1 + · · ·+ fn, where fj ∈ Aj, for j = 1, . . . , n. Therefore

∀ i = 1, . . . k (1 + f1 + · · ·+ fn)mi = 0; hence mi = 0.

So, M = 0. The proof is complete.

The following is a graded analogue of the corresponding ungraded result.


Lemma 4.3.8. LetA = A0⊕A1⊕A2⊕· · · be a graded commutative ring and P =⊕

n∈Z Pn,Q =

⊕n∈ZQn be a finitely generated graded projective A-modules. Let A+ =

⊕i≥1Ai. Let

ϕ0 : PA+P

∼−→ QA+Q

be an isomprphism of A0-modules. Then, ϕ0 lifts to and isomorphismϕ : P

∼−→ Q of graded A-moduels.

Proof. Similar to the proof of the corresponding ungraded result, using (4.3.7).

Example 4.3.9. Let A = A0⊕A1⊕A2⊕ · · · be a graded commutative ring and Pgr(A)

be the category of finitely generated graded projective modules P =⊕

n∈Z Pn.

1. Then, for all integers i ≥ 0, the groups Ki (Pgr(A)) is a Z[t, t−1]-module, wheremultiplication by t is induced by the functor

Pgr(A) −→Pgr(A) given by P 7→ P (−1), where P (−1)n = Pn−1

Proposition 4.3.10. For all integers i ≥ 0, there is a Z[t, t−1]-module isomorphism

Z[t, t−1]⊗Z| Ki(A0)∼−→ Ki (Pgr(A)) 1⊗ x 7→ (A⊗A0 −)∗x

Proof. Note the functor

P(A0) −→Pgr(A) given by Q 7→ A⊗A0 Q

is an exact functor. So, it induces a the following homomorphisms:

Ki(A0) //

Ki (Pgr(A))

Z[t, t−1]⊗Z Ki(A0)

55

Given P in Pgr(A) and integer k, let

Fk(P ) = A

(⊕n≤k

Pn

)⊆ P

and

Pq = P ∈ Obj (Pgr(A)) : F−q−1(P ) = 0, Fq(P ) = P be the full subcategory.

We have a functor

T : Pgr(A) −→Pgr(A0), T (P ) = A0 ⊗A P =P

A+P, where A+ = A1 ⊕ A2 ⊕ · · ·


It follows from Lemma 4.3.8, that P is non-canonically isomorphic to

A⊗A0 T (P ) =∐n

A(−n)⊗A0 T (P )n (4.14)

Now, it follows from (4.14):

1. Fk : Pgr(A) −→Pgr(A) is an exact functor. This is because

Fk(P ) =∐n≤k

A(−n)⊗A0 T (P )n

2. For each q, there is a filtration of the functors 1Pq :

0 = F−q−1 ⊆ F−q ⊆ · · · ⊆ Fq = 1Pq

(here Fi means the restriction of Fi.

3. Fk(P )Fk−1(P )

∼= A(−k)⊗A0 T (P )n.

Define the mapΦ :∐−q≤n≤q t

n ⊗Ki(A0) −→ Ki(Pq) tn ⊗ x 7→ (A(−n)⊗A0 −)∗x

Ψ : Ki(Pq) −→∐−q≤n≤q t

n ⊗Ki(A0) x 7→ ⊕−q≤n≤qtn ⊗ (Tn)∗(x)

By Corollary 4.3.5, one can see Φ and Ψ are inverse of each other. Since Pgr(A) is unionof Pq, the proposition follows. The proof is complete. .

4.4 Reduction by Resolution

From the point of view of the Classical K-Theory, the following example motivates thissection.

Example 4.4.1. Let A be a noetherian commutative ring. Let P(A) be the category offinitely generated projective A-modules and let H(A) be the category of finitely generatedA-modules, with finite projective dimension. Then, K0(P(A)) ∼= K0(H(A)).(Use Definition 4.2.5 only.)

Proof. For definitions of the classical K0, refer to Definition 4.2.5 (4) . As usual, theinclusion functor ι : P(A) −→ H(A) induces an homomorphism K0(ι) : K0(P(A)) ∼=K0(H(A)). Now, we define the inverse of this map K0(ι).

Let H denote the isomorphism classes objects in H(A). For an object inM in H(A), letM denote its isomorphism class in H and [M ] denote its class in K0(H(A)) (and likewisefor P(A)). For such an M , let

0 // Pn // Pn−1// · · · // P1

// P0//M // 0

4.4. REDUCTION BY RESOLUTION 131

be a P-resolution of M . We say that P• is a P(A)-resolution of M . Now, define

ϕ0 : H −→ K0(P(A)) by ϕ(M) =n∑i=0

(−1)i[Pi].

We need to show that ϕ0 is well defined. Let Q• be another P(A)-resolution of M (byaugmenting zeros at the tails, we can assume that both P• and Q• have same length n). Itwould be sufficient to prove that there is an exact sequence

· · · // P1 ⊕Q2// P0 ⊕Q1

// Q0// 0

To see this, note that by properties of projective modules, there is a map of chain complexesf• : P• −→ Q•, as in the following commutative diagram:

0 // Pn //

fn

Pn−1//

fn−1

· · · // P1//

f1

P0//

f0

M // 0

0 // Qn// Qn−1

// · · · // Q1// Q0

//M // 0

Let C(f•) denote the cone of f• (see [W, pp. 18-19]). So, C(f•)n = Qn ⊕ Pn−1 and thesequence

0 // Q• // C(f•) // P•[−1] // 0

This gives rise to a long exact sequence of homologies

· · · // H1(P•) // H1(Q•) // H1(C(f•)) // H0(P•) // H0(Q•) // H0(C(f•)) // 0

Since H0(P•) = H0(Q•) = M , and Hi(P•) = Hi(0•) = 0, ∀ i ≥ 1, it follows Hi(C(f•)) =0 ∀ i. This establishes the above and hence ϕ0 is an well defined set theocratic map. So,ϕ0 extends to a homomorphism, as follows

H ϕ0 // _

K0(P(A))

Z(H)

ϕ0

88

Now, K0(H(A)) = Z(H)R(H)

. We show that ϕ0 respects the relations R(H). To see this,suppose

0 // K //M // C // 0 is exact in H(A).

Given finite P(A)-resolutions P• of K and Q• of C (say of same length n), inductively,one can build a resolution

0 // Qn ⊕ Pn // Qn−1 ⊕ Pn−1// · · · // Q1 ⊕ P1

// Q0 ⊕ P0//M // 0

Thereforeϕ0(M) = ϕ0(K) + ϕ0(C)


Hence, ϕ0 factors through a homomorphism:

H ϕ0 // _

K0(P(A))

Z(H)

ϕ0

88

// K0(H(A))

ϕ

OO

It is easy to see that K0(ι) and ϕ are inverse of each other. The proof is complete.

4.4.1 Extension closed and Resolving Subcategories

Definition 4.4.2. Suppose E is an exact category (with a set of isomorphism classes).For subsequent reference, let

0 // K //M // C // 0 be an exact sequence in E . (4.15)

Recall, short exact sequences (4.15) is also known as "extensions" (see [HS, Chapter III]),or "conflations". A full subcategory P of E is called an extension closed subcategoryof E ,

1. if P has a zero object,

2. and, if for any exact sequence (4.15), whenever K,C are isomorphic to some objectsin P, then so is M .

(see Definition 1.3.3). We have the following:

1. If P is an extension closed subcategory of E , then P is also an exact category, wherea sequence in P is declared exact, if and only if it is exact in E . Consequently,P → E is an exact functor.

2. Further, if P is an extension closed subcategory of E , then Q(P) is a subcategoryof Q(E ), while not necessarily a full subcategory.

The Resolution Theorem

Theorem 4.4.3 (Theorem 3). Suppose P is a full subcategory of an exact category E ,which is extension closed. Assume

1. Given a short exact sequence (4.15) in E , if M ∈ Obj(P), then K ∈ Obj(P).


2. For any C ∈ Obj(E ), there is a short exact sequence (4.15), with M ∈ Obj(P).

(Equivalently, (1) P is closed under subobjects and (2) any C ∈ Obj(E ) is subquotient ofan object in P.)

Then, the inclusion QP → QE is a homotopy equivalence. Consequently, Ki(P)∼−→

Ki(E ) ∀ i ≥ 0.

Proof. Let C be the full subcategory of Q(E ), whose objects are those of P. Recall,Q(P) is not necessarily a full subcategory of Q(E ). However, we have the factorization

Q(P)g //

$$

C

f

Q(E )

We prove that both f, g are homotopy equivalence. First, we show g is a homotopy equiv-alence. We do this, we use Theorem 4.1.22, and show that ∀ P ∈ Obj(P), the categoryg \ P is contractible.

Fix P ∈ Obj(P). Objects in g \ P and given by (C, u), where C ∈ Obj(P) andu : C −→ P is a morphism in QE , which is given by C M

poooo // P . In terms ofadmissible layers, it is given by the following diagram: Let J be the ordered set (category)of E -admissible layers (K,M) in P , such that M/K ∈ Obj(P). Diagramatically:

0 // K //Mp // _

C // 0

P

horizontal line is a conflation in E , M/K ∼= C ∈ Obj(P),

and M → P is an inflation.

To describe the morphisms in J , let (K,M), (K ′,M ′) in J . We have M/K,M ′/K ′ ∈Obj(P). Define

(K,M) ≤ (K ′,M ′) if

M ′/M

K ′ // _

M ′

OOOO

K //

M?

OO

K/K ′

with K/K ′,M/M ′ ∈ Obj(P)

That means, the vertical sequences are conflations in E , to start with.


So, (J ,≤) is an ordered set (category). Define a functor

Φ : J −→ g/P by Φ((K,M)) =(M/K M

//oooo P)

Also:

Φ((K,M) ≤ (K,M ′)) =

M/K M //oooo P

M/K ′ _

OOOO

M ′/K ′ M ′ //oooo P

This diagram of three maps in QE commute and the vertical map is in QP. We have:

1. Clearly, Φ is essentially surjective.

2. Consider

Φ : Mor((K,M), (K,M ′)) = ≤ −→Mor(Φ(K,M),Φ(K ′,M ′))

Since, the domain is a singleton, Φ is injective. We show thatMor(Φ(K,M),Φ(K ′,M ′))is a singleton. Consider a map in g/P :

M/K M //oooo P

C _

OOOO

M ′/K ′ M ′ //oooo P

The composition of the vertical with the bottom map is given by:

W

//M ′ //

P

M/K Coooo //M ′/K ′

The commutativity =⇒

M/K M //oooo

o τ

P

M/K W //oooo P

Identifying M ∼−→ W , we have the commutative diagram

M

f

//M ′ //

g

P

M/K Coooo //M ′/K ′

It follows from properties of pullback, that ker(f) = ker(g) = K ′ (see [HS, II.Thm6.2]). So, C ∼= M/K ′. Therefore Mor(Φ(K,M),Φ(K ′,M ′)) is a singleton.


This establishes that J −→ g/P is an equivalence of categories. So, we will now provethat J is contractible.

So, in J , we have

(K,M) ≤ (0,M) ≥ (0, 0). Define

F : J −→ J by F (K,M) = (0,M)C : J −→ J by C(K,M) = (0, 0)

We define two natural transformations:τ1 : IDJ −→ F by (K,M) ≤ (0,M)τ2 : C −→ F by (0, 0) ≤ (0,M)

By Proposition 4.1.9, there are homotopy: IDBQJ 7→ BF and Constant 7→ BF . So, IDBQJis homotopic to the constant map. Hence J is contractible. Therefore g/P is contactable.By Theorem 4.1.22, g is a homotopy equivalence.

Now, we prove that f : C −→ QE is a homotopy equivalence. To do this, we will provethat, for M ∈ Obj(QE ) = Obj(E ), M/f is contractible. Recall, the objects (P, u) of M/fare all maps u : M −→ f(P ) = P in QE , (where P ∈ Obj(P)) which looks like

M Qjoooo

i// P in E

Write F = M/f and F ′ = (P, u) ∈ Obj(F ) : u surjective be the full subcategory.

We proceed to prove that the inclusion functor F ′ −→ F is a homotopy equivalence. Hereare some observations:

1. For an object (P, u) in F ′, P is in P.

2. It follows, that a morphism θ : X −→ Y in F ′ is surjective:

X :

θ

M Qjoooo

iQ

X : M Pjoooo

iP

j

OOOO

3. Given X = (P, u) ∈ Obj(F ), we have u = i!j! an in M Q

joooo

i// P .

By hypothesis (1), Q ∈ Obj(P). So, X := (Q, j!) ∈ F ′. Now, ι := i! : X −→ X inQE is a map in F , as given by the diagram:

Xι

M Qjoooo Q

i!

X M Qjoooo

i// P


We claim ι : X −→ X is an universal arrow from an object of F ′ to X. This means, givenand arrow α : X ′ −→ X, with X ′ ∈ F ′, there is an unique map β : X ′ −→ X , such thatthe diagram

X ′α

βX // X

commute.

To see this let X ′ = (Q′, j′) and let α : X ′ −→ X be given by the commutative diagram inQE :

X ′

α

M Q′j′oooo Q′

W

j”

OOOO

_

i′

X M Q

joooo

i// P

=⇒M W

τo

j′j”oooo i′ // P

M Qjoooo

i// P

=⇒ τ : W∼−→ Q, i′ = iτ

We can assume W = Q and i = i′. It follows, Q′ is a subquotient j” : Q Q′ and

X ′α

(j”)!

X ι

// X

commutes. This establishes existence of β.

The uniqueness of the vertical map follows from the fact that, decomposition of α, ascomposition of injective and surjective maps, is unique: Suppose

X ′α

βX ι

// X

Write

X ′

β

M Q′oooo Q′

W

OOOO

_

X M Qoooo Q

=⇒ W = Q

and

X ′

β

M Q′j′oooo Q′

Q

ζ

OOOO

X M Qjoooo Q


Also,

X ′

β

M Q′j′oooo Q′

Q

ζ

OOOO

Xι

M Qjoooo Q _

i

X M Qjoooo // P

So, β is given by ζ !. Now, ιβ = α means that the vertical map coincide with the verticalmap in the diagram of α, which is given by

Q′ Qζoooo i //

ρo

P

Q′ Qj”oooo

i// P

=⇒ ρ = Id

Therefore the (j”)! = ζ !. This completes the proof of uniqueness.

So, it is established that X −→ X as an universal arrow, from an object of F ′. Wedenote the functor F : F −→ F ′, i. e. F (X) = X . In fact, universality, establishes that Fis a functor. This also implies that F is a right adjoint of the inclusion functor F ′ −→ F :

∀ U ∈ Obj(F ′), X ∈ Obj(F) there is a bijection MorF ′(U , F (X))∼−→MorF(U , X)

By Corollary 4.1.10, it follows that the inclusion functor F ′ −→ F is a homotopy equiva-lence.

So, it would be enough to prove that F ′ is contractible. We will prove that (F ′)op iscontractible. Objects in (F ′)op are deflations P M , with P ∈ Obj(P) and morphisms

are commutative diagrams

P // //

M

Q

>> >>

. Fix a deflation q0 : P0 M , with P0 ∈ Obj(P),

given by hypothesis (2). Given another object q : P M in (F ′)op, we have the pull back

P ×M P0p1 // //

p2

P0

q0

P q// //M

Since ker(p2) = ker(q0), and since it is in Obj(P) by hypothesis (1), it follows from theleft vertical line, that P ×M P0 ∈ Obj(P). We define functors:

Id : (F ′)op −→ (F ′)op Id(P M) = (P M)F : (F ′)op −→ (F ′)op F (P M) = (P ×M P0 M)C : (F ′)op −→ (F ′)op C(P M) = (P0 M)


So, p2 : F −→ Id and q0p1 : F −→ C are natural transformations. So, by Proposition 4.1.9,there are homotopies: BF 7→ IDBF ′ and BF 7→ Constant. Therefore,F ′ is contractible.


The hypothesis of the following (4.4.4) is slightly weaker than being a resolving sub-category.

Corollary 4.4.4. Suppose E is an exact category and P is an extension closed full sub-category. For subsequent reference, let

0 // K //M // C // 0 be a conflation in E . (4.16)

Assume

1. For any conflation (4.16) in E , if C,M ∈ Obj(P), then K ∈ Obj(P).

2. Given a deflation j : M P in E , with P ∈ Obj(P),

∃ defletions g : P ′ P, f : P ′ M 3 P ′ ∈ Obj(P) and

P ′

g

f

~~~~M

j// // P

commute.

(With P = 0, any object M in E is surjective image of an object P ′ ∈ Obj(P).)

For integers n ≥ 0, let Pn denote the full subcategory of objects M E , that has a P-resolution of length ≤ n. (Here the concept of "resolutions" needs to be understood, interms of deflations and inflations.) Then,

1. Pn is an extension closed full subcategory of E . Therefore, Pn is an exact category.

2. Write P∞ =⋃

Pn. Then, P∞ is also an exact subcategory of E .

3. Finally,Ki (P)

∼−→ Ki (Pn)∼−→ Ki (P∞) ∀ i ≥ 0, n ≥ 0.

Proof. The proof follows by an application of Theorem 3 (4.4.3), to the inclusion Pn ⊆Pn+1, with additional help from the following Lemma 4.4.5. Note, since P ⊆Pn, any C ∈Obj(Pn+1) is a subquotient of an object in Pn. Further, Pn is closed under subobjects,in Pn+1, by Assertion 2 of Lemma 4.4.5. So, Theorem 3 (4.4.3) applies to the inclusionPn ⊆Pn+1. The proof is complete.


Lemma 4.4.5. Suppose P and E and for integers n ≥ 0, Pn be as in (4.4.4). Suppose

0 // Ki //M

j // C // 0 be exact in E . (4.17)

Under the hypothesis of (4.4.4), and for integers n ≥ 0, we have

1. If K,C ∈ Obj(Pn+1), then M ∈ Obj(Pn+1).(Therefore, Pn is extension closed in E ).

2. If M ∈ Obj(Pn) and C ∈ Obj(Pn+1), then K ∈ Obj(Pn).

3. If M,C ∈ Obj(Pn+1), then K ∈ Obj(Pn+1).

Proof. (The proof is imitation of the standard arguments in commutative algebra, regardingprojective dimension of modules M , over commutative noetherian rings A. For example,see [Mat, Lemma 2, pp. 128].) We prove the case n = 0, 1 and then use induction.

First we prove (1), for n = 0, which is to prove that P1 is extension closed in E . LetConsider the sequence (4.17). Assume K,C ∈P1. Consider the diagrams:

0

0

Q1

P1

Q0

∂

P0

d

η

~~K

i//

Mj// C

0 0

=⇒

0

0

0 // Q1

// N

// P1

// 0

0 // Q0

∂

// Q0 ⊕ P0//

δ=(i∂+η)

// P0

d

// 0

0 // K //

M // C

// 0

0 0

In the left hand diagram, two vertical lines provide P-resolutions of K and C. The mapη is obtained by hypothesis (2) of (4.4.4), with jη = d. The second commutative diagramextends the first one, where δ is induced by ∂ and d.

Recall, by (1.3.4) E can be embedded as a fully exact, extension closed subcategory ofan abelian category C . Let N = ker(δ) in C . By Snake Lemma 1.5.4,

Q1// N // P1

// coKer(∂) // coKer(δ) // coKer(d)

is exact. Since coKer(∂) = coKer(d) = 0, it follows coKer(δ) = 0. Further,

Q1// N // P1

// 0 is exact in C . Also, 0 // Q1// N is exact in C .

Therefore,0 // Q1

// N // P1// 0 is exact in C .


Since P → E → C are extension closed, N ∈ Obj(P). Now, the middle vertical line isa conflation is C . Since the E → C is fully exact, the middle vertical line is conflationin E . Therefore, M ∈ Obj(P1). This completes the proof that P1 is an extension closedsubcategory of E .

Now, we prove (2), for n = 0. So, M ∈ Obj(P0) and C ∈ Obj(P1). Consider thediagram:

P1

P1

g

0 // K // P1 ×C M //

P0

f

// // 0

0 // Ki

//Mj

// C // 0

where the last vertical line is a resolution of C, with P0, P1 ∈ Obj(P) and P1 ×C M isthe pullback of (j, f). If follows from (1.5.1) that the middle vertical line and as well themiddle horizontal line are conflesions, in E . By extension closed property of P it followsthat P1 ×C M ∈ Obj(P). Now, by hypothesis (1) of Corollary 4.4.4, K ∈ Obj(P). Thisestablishes (2) for n = 0.

Now, we prove (3), for n = 0. So, letM,C ∈ Obj(P1). Consider the pullback diagram:

P1 _

P1 _

// // 0 _

P0 ×M K

// P0

d

jd // // C

K

i//M

j// // C

It follows from Snake Lemma 1.5.4, that the middle horizontal line is a conflesion. SinceC ∈ Obj(P1) and P0 ∈ Obj(P0), by Case n = 0 of (2), K ∈ Obj(P1).

So, the lemma is established in the case n = 0. Now, the proof is completed byinduction.

Resolving Subcategories

Definition 4.4.6. Suppose E is an exact category (with a set of isomorphism classes).For subsequent reference, let

0 // K //M // C // 0 be an exact sequence in E . (4.18)

Now suppose P is an extension closed subcategory of E , (hence an exact subcategory).Then, P is said to be a resolving subcategory of E , if the following two conditions aresatisfied:


1. Given an exact sequence (4.18) in E , if M,C are in P, then so is K.

2. For all objects M in E , there is an exact sequence:

0 // Pn // Pn−1// · · · // P1

// P0//M // 0

with Pi in P. In this case, we say P• is a P-resolution of M , using deflations.

Example 4.4.7. Suppose A is noetherian commutative ring.

1. Note, P(A) is an exact subcategory of fMod(A) = Coh(A), it is not a resolvingsubcategory of fMod(A). However, if A is a regular ring, then P(A) is a resolvingsubcategory of fMod(A) = Coh(A).

2. Let H(A) be the full subcategory of fMod(A) whose objects are the A-modules Min fMod(A), with finite projective dimension. Then, H(A) is an exact subcategoryof fMod(A). Further, P(A) is a resolving subcategory of H(A).

Theorem 4.4.8. Let E be an exact category and P ⊆ E be a resolving subcategory. Then,for all integers, i ≥ 0, we have Ki(P)

∼−→ Ki(E ) is an isomorphism.

Proof. We apply Corollary 4.4.4. The first condition of (4.4.4) follows from definitionof resolving subcategory. Now suppose j : M P in E is a deflation in E , with P ∈Obj(P). Again by definition of resolving category, there is a deflation ι : P0 M , withP0 ∈ Obj(P). Therefore, the diagram of deflations

P0

ι~~~~jι

Mj// // P

commute.

So, the second condition of (4.4.4) is satisfied. By the second condition in the definition ofresolving category, P∞ = E . So, the proof is complete by Corollary 4.4.4.

The following is extension of the classical Example 4.4.1.

Corollary 4.4.9 (Corollary 2). Let A be a commutative noetherian ring and P(A) ⊆ H(A)

be as in Example 4.4.7. Then, for all integers i ≥ 0, Ki(P(A))∼−→ Ki(H(A)) is an

isomorphism. In particular, if A is a regular ring, then Ki(P(A))∼−→ Ki(Coh(A)).

Proof. Follows from Corollary 4.4.8. We comment that if dimA = d thenH(A) = (P(A))d(see [Mat]).


Corollary 4.4.10 (Corollary 3). Let T = Ti : i ≥ 1 be an exact connected sequence offunctors from an exact category E to an abelian category A , meaning

1. Ti : E −→ A are functors (not necessarily exact),

2. For each conflation

K //M // // C in E (4.19)

and i ≥ 2, there is a morphisms δi : Ti(C) −→ Ti(K) such that following

· · · // T2(K) // T2(M) // T2(C)δ2 // T1(K) // T1(M) // T1(C)

is exact.

Let

P = M ∈ Obj(E ) : Ti(M) = 0 ∀ i ≥ 1 be the full subcategroy E .

Then, P is an extension closed subcategory of E .

Further assume that

1. for all M ∈ Obj(E ) there is a deflation P M for some P ∈ Obj(P).

2. for all M ∈ Obj(E ), Tn(M) = 0 ∀ n 0.

Then, Ki(P)∼−→ Ki(E ) ∀ i ≥ 0.

Proof. It follows immediately, that P is extension closed and hence Ki(P) are defined.For integers n ≥ 0, define the full subcategories

Pn = M ∈ Obj(E ) : Ti(M) = 0 ∀ i ≥ n+ 1

It follows, that Pn ⊆ Pn+1 is an extension closed subcategory, for all n ≥ 0. It follows,given a conflation (4.19) in Pn+1, ifM ∈ Obj(Pn) andK ∈ Obj(Pn). Both the conditionsof Theorem 3 (4.4.3) are satisfied. Hence, for all integers i ≥ 0, Ki(Pn)

∼−→ Ki(Pn+1) areisomorphisms.

By hypothesis, E = P∞ = ∪n≥0Pn. Therefore, Ki(E ) = Ki(P0) for all i ≥ 0. Theproof is complete.

Remark 4.4.11. Compare the definition of "connected sequence of fuctors" with that ofδ-functors in [Hr, Section III.1, pp. 205].

4.5. DÉVISSAGE AND LOCALIZATION IN ABELIAN CATEGORIES 143

4.5 Dévissage and Localization in Abelian Categories

We would be working on the following set up.

Definition 4.5.1. Let A be an abelian category, with an set isomorphism classes ofobjects. Let B ⊆ A be a subcategory, such that

1. B ⊆ A is full subcategory,

2. B is closed under subobjects, quotient objects and finite product in A .

Note, in this case,

1. B is an abelian category and the inclusion B → A is an exact functor.

2. Further, QB → QA is a full subcategory.

The following is the well known Dévissage Theorem.

Theorem 4.5.2 (Dévissage). Let B → A be as in the set up (4.5.1). Further assume,every object M ∈ Obj(A ) has a filtration:

0 = M0 →M1 →M2 → · · · →Mn =: M 3 ∀ j = 1, . . . , n− 1Mj

Mj−1

∈ Obj(B).

Then, QB → QA is a homotopy equivalence and hence Ki(B)∼−→ Ki(A ) is an isomor-

phism, for all integers i ≥ 0.

Proof. Let f : QB → QA be the inclusion functor. Fix M ∈ Obj(A ). By Theorem A(4.1.22), it suffices to show that f/M is contractible. An object of f/M is given by a pair(N, u), where N ∈ Obj(B) and u ∈ MorQA (N,M). Such a morphism u is given by anA -admissible layer (M0,M1) in M , as in the diagram:

M0 _

M1

p

ι //M

N

with M0 = ker(p).

Let J (M) be a category (ordered) of all such A -admissible layer (M0,M1) in M , withM1/M0 ∈ Obj(B). Further, (as in the proof of Theorem 4.4.3)

(M0,M1) ≤ (M ′0,M

′1) in M ′

0 ≤M0 ≤M1 ≤M ′1


We clarify, diagrammatically:

M ′0

M ′0

M1

π

q

~~

//M ′1

p

//M

N Woooo // N ′

with M ′0 = ker(p), M0 = ker(q).

Note, W ∼= M1

M ′0⊆ M ′0

M ′0. Hence W ∈ Obj(B).

Since J (0) is trivial and M has a filtration, with quotient in B, it is enough to provethat that inclusion ι : J (M ′) → J (M) is a homotopy equivalence, with M ′ ⊆ M andMM ′∈ Obj(B. We define two functors:

r : J (M) −→ J (M ′) (M0,M1) 7→ (M0 ∩M ′,M1 ∩M ′)s : J (M) −→ J (M) (M0,M1) 7→ (M0 ∩M ′,M1)

These are well defined because

M1 ∩M ′

M0 ∩M ′ ⊆M1

M0 ∩M ′ ⊆M1

M0

× M1

M ′ ∈ Obj(B).

Consider the following:

1. rι = IdJ (M ′).

2. There are natural transformations:ιr −→ s (M0 ∩M ′,M1 ∩M ′) ≤ (M0 ∩M ′,M1)IdJ (M) −→ s (M0,M1) ≤ (M0 ∩M ′,M1)

Now, by Proposition 2 (4.1.9), r is a homotopy inverse of ι. The proof is complete.

The Nilpotent idealsHere is an important corollary of our primary interest.

Lemma 4.5.3. Suppose A is a noetherian ring (commutative) and I is a nilpotent ideal.Then, for all integers i ≥ 0, we have Ki

(fMod

(AI

)) ∼−→ Ki(fMod(A)).

Proof. Apply the Dévissage Theorem (4.5.2), to the inclusion fMod(AI

)→ fMod(A).

Note In = 0 for some integer n ≥ 0. So, for any objectM in fMod(A), there is a filtration:

0 → In−1M → · · · IM →M.



4.5.1 Semisimple Objects

Definition 4.5.4. Let A be an abelian category.

1. An object M in A is called a simple object if M has precisely two subjects, namely,0 6= M .

2. An objectM in A is called a semisimple object if there is a surjective maps ⊕i∈ISi M , where Si :∈ I is a set of simple objects.

3. We say and object M has finite length, if M has a finite filtration

0 = M0 →M1 →M2 → · · · →Mn =: M 3 ∀ j = 1, . . . , n− 1Mj

Mj−1

is simple.

(In this case, define length `(M) := n is well defined. It needs a proof that it is welldefined.)

Lemma 4.5.5. Suppose A is an abelian category andM be an object. Then, the followingare equivalent:

1. M is semi simple.

2. M ∼= ⊕i∈ISi, where Si :∈ I is a set of simple objects. (So, if `(M) = Card(I),)

Proof. (2) =⇒ (1) is obvious. To prove ((1) =⇒ 2), let f : ⊕i∈ISi M , be a surjectivemal, where Si :∈ I is a set of simple objects. Let

I := I0 ⊆ I : f|∪i∈I0Si is injective

By Zorn’s lemma, I has a maximal element J ⊆ I. Write N = ∪i∈JSi. Claim f0 := f|N :

N∼−→ M is an isomorphism. If not, there is i1 ∈ I \ J such that f(Si1) 6⊆ g(N). Let

f1 := f|Si1 : Si1 −→M . We have,

1. f0 : N −→M is injective and f1 : S1 −→M is injective.

2. Further, f|(N⊕Si1 ) : N ⊕ Si1 = f0 ⊕ f1.

3. Since N ⊕ Si1 ∼= N × Si1 , it follows that f|(N⊕Si1 ) is injective. This is a contradictionto the maximality of J .

Therefore, f0 is an isomorphism. The proof is complete.


Corollary 4.5.6. Let A be an abelian category (with a set of isomorphism classes ofobjects), such that every object has finite length. Then,

∀ i ≥ 0 Ki(A) =∐j∈J

Ki(Dj) recall∐

= ⊕

where Xj : j ∈ J is a set of representative for isomorphism classes of simple objects ofA and Dj = End(Xj)

o. (Note, Dj is a devision ring.)

Proof. Let B be the full subcategory of semi simple objects in A . Then, by DévissageTheorem (4.5.2), Ki(B)

∼−→ Ki(A ) is an isomorphism, ∀ i ≥ 0.

Let Bj be the full subcategory of B, only one simple object Xj (up to isomorphism).It follows B is limit of finite products

∏mk=1 Bjk . Since Ki commute with finite product

and filtered limits, if follows

∀ i ≥ 0 Ki(B)∼−→⊕j∈J

Ki(Bj) is an isomorphism.

Now, the proof follows from the following lemma (4.5.7).

Lemma 4.5.7. Suppose B is an abelian category such that (1) each object has finitelength, and (2) B has only a single simple object X, up to isomorphism. Let D :=

Mor(X,X)o and P(D) denote the category of finite dimensional left D-vector spaces.The functor

F : B −→P(D) 3 F (M) = MorB(X,M)

is an equivalence of categories.

Proof. It follows from Lemma 4.5.5 that, each object M ∼= Xn for some integer n ≥ 0.

4.5.2 Quillen’s Localization Theorem

Theorem 4.5.8 (Localization Theorem). Let A be an abelian category and B ⊆ A be aSerre subcategory (1.4.5). Let A

Bbe the quotient abelian category. Let ι : B → A be the

inclusion functor and q : A −→ AB

the canonical functor. Then, there is an exact sequence

· · · q∗ // K1

(AB

)// K0 (B)

ι∗ // K0 (A )q∗ // K0

(AB

)// 0

Proof. Fix zero 0 in A and let 0 denotes its image in AB. From the defintion of Serre

subcategories (1.4.5) and the construction of the quotient AB, it follows, B is the full


subcategory of objects M in A such that qM ' 0.Consider the composition:

QBQι //

0 ""

QA

Qq

QAB

up to natural equivalence.

Consider 0 \ Qq. Since MorA (0,M) and MorA (M, 0) are singletons, objects in . SinceMorA

B(0,M) and MorA

B(M, 0) are singletons, objects in 0 \ Qq are given by, equivalence

classes of,

0 0 i //oooo M in Q

A

B, i ∈ A

B.

If M ∈ Obj(B) then i is an isomorphism in AB. We consider two functors

F : QB −→ 0 \Qq F (M) = (M, 0∼−→ qM)

G : 0 \Qq −→ QA G(N, u) = N

Further, there we have the commutative diagram:

QB F //

Qι ##

0 \QqG

QA

The following would settle of proof of this theorem:

1. We would apply Theorem B (4.1.32) to Qq.To apply Theorem B (4.1.32) to Qq, we would prove that for all u : V ′ −→ V in QA

B,

the map u∗ : V \Qq −→ V ′ \Qq is a homotopy equivalence.Recall, u∗ sends (W, v) 7→ (W, vu).Since every morhism in QA

Bfactors in to "surjective" and "injective", and also by

reversing the arrows, we can assume u is injective.It is claimed i!(iV ′)! = (iV )!, which I do not see.So, it is enough to prove this for injective maps (iV )! for V in A

B.

2. Also, we prove that F : QB −→ 0 \Qq is a homotopy equivalence.Let FV be the full subcategory of V \ Qq, consisting of pairs (M,u) such thatu : V

∼−→ qM is an isomorphism. In particualar, F0 → 0 \ Qq. It is clear F0

is equivalent to B. Hence, this point follows form the following Lemma 4.5.9.

Lemma 4.5.9. The inclusion functor FV → V \Qq is a homotopy equivalence.

Proof. For the purposes of this course, we skip the proof. In stead, we do other thingsduring the remaining four lectures.


Remark 4.5.10. We summarize application of the above to K-Theory of schemes.Given a scheme X and a closed subscheme Z ⊆ X, the the localization theorem (4.5.8)applies to the inclusion Coh(Z) ⊆ Coh(X), as follows:

1. Let CohZ(X) ⊆ Coh(X) be the full subcategory of coherent OX-modules, F suchthat Supp(F) ⊆ Z. Obviously, i : CohZ(X) ⊆ Coh(X) is an exact functor. There-fore, i induce a map of the K-theories of these two exact categories.

2. Let U := X \ Z. Then, the restriction functor j : Coh(X) −→ Coh(U) is an exactfunctor. Therefore, j induce a map of the K-theories of these two exact categories.

3. It follows, Coh(X)CohZ(X)

∼−→ Coh(U) is an equivalence of categories.

4. By localization theorem (4.5.8) (although I did not state it inthis way):

BQCohZ(X) // BQCoh(X) // BQCoh(U)

is a homotopy fibration of topological spaces (in fact CW-complexes). Further, byDévissage Theorem (4.5.2), BQCohZ(X)

∼−→ BQCoh(Z) is a homotopy equivalence.Putting these two fcts together,

BQCoh(Z) // BQCoh(X) // BQCoh(U)

is a homotopy fibration of topological spaces (in fact CW-complexes).

5. This homotopy fibration leads to long exact sequence of K-groups. With notations

Gi(X) := Ki(Coh(X)) := πi+1(BQCoh(X))

there is a long exact sequence

· · · // G1(Z) // G1(X) // G1(U)∂1 // G0(Z) // G0(X) // G0(U) // 0

6. Only when X is regular (noetherian and separated) (hence so is U), we can sayKi(X)

∼−→ Gi(X) are , and as well Ki(U)∼−→ Gi(U). However, unless Z is also

regular, we cannot say that Ki(Z)∼−→ Gi(Z).

7. Finally, there is no satisfactory analogue of the Localization Theorem (4.5.8) thatapplies to V ect(X)

4.6. K-THEORY SPACES AND REFORMULATIONS 149

4.6 K-Theory Spaces and Reformulations

As was seen above, the K-groups Ki(E ) of an exact category was determined by theclassifying spaces BQE . From this perspective, it may make more sense to define K-Theory as a topological space, as follows.

Definition 4.6.1. Let E be an exact category. The K-theory space of |SE is defines tobe the pointed topological space

K(E ) := Ω (BQE ) the loop space of BQE , at 0

with the base point the constant loop (see Definition 2.4.11). Therefore, the K-groups aregiven by

Ki(E ) = πi+1 (BQE , 0) = πi (K(E ), 0) ∀ i ≥ 0.

So, most of the above can be reformulated in terms of the K-Theory spaces.

The Resolution Theorem is reformulated, as follows.

Theorem 4.6.2. Let E be an exact category and P ⊆ E be a resolving subcategory. Then,the map

K(P) −→ K(E ) is a homotopy equivalence.

Theorem 4.6.3 (Dévissage). Under the hypotheses of Theorem 4.5.2, the map

K(B) −→ K(B) is a homotopy equivalence.

Theorem 4.6.4 (Localization). Under the hypotheses of the Localization Theorem (4.5.8)

K(B) //K(A ) //K(

AB

)is a homotopy fibration.

4.7 Negative K-Theory

With my primary interest on P(A), where A is a commutative noetherian ring, I cannotpinpoint what was the need for negativeK-Theory. However, failure of applicability of bothDévissage Theorem (4.5.2) and Localization Theorem (4.5.8) , to K-Theory of projectivemodules, leaves a taste of incompleteness. At this time, I would presume that that wouldbe a justification for Negative K-Theory. We follow the papers of Schlichting ([Sm1, Sm2]).Before we introduce negative K-theory, we develop some background.


4.7.1 Spectra and Negative Homotopy Groups

We begin with the related topological background on (negative) homotopy groups of spetra.

Definition 4.7.1. A Spectrum of pointed topological spaces is a sequence of topologicalspaces

E0, E1, E2, . . . , together with maps σk : Ei −→ ΩEk+1 to be called bonding or structure maps.

Such a Spectrum, would be denoted by (E, σ).

Fix an integer n ∈ Z. To define πn(E), fix integers k ≥ l ≥ −n. So, iteratively, we havemaps

Ωk−lEkΩk−lσk // Ωk−l+1Ek+1

Ωk−l+1σk+1 // Ωk−l+2Ek+2// · · ·

Define

πn(E) = co lim

(πn+l

(Ωk−lEk

) Ωk−lσk // πn+l

(Ωk−l+1Ek+1

) Ωk−l+1σk+1 // πn+l

(Ωk−l+2Ek+2

)// · · ·

)1. One checks that this definition does not depend of the choices on l, k, as long ask ≥ l ≥ −n.

2. If n ≥ 0 then, we can take k = l = 0. In this case, we have

πn(E) = co lim(πn (E0)

σ0 // πn (Ω1E1)Ω1σ1 // πn (Ω2E2) // · · ·

)We can also take k = l = 1, in which case

πn(E) = co lim(πn+1 (E1)

σ1 // πn+1 (Ω1E2)Ω1σ2 // πn+1 (Ω2E3) // · · ·

)We can also take k = 2, l = 1, in which case

πn(E) = co lim

(πn+1 (Ω1Ek)

Ω1σk // πn+1 (Ω2E3)Ω2σ3 // πn+l

(Ωk−l+2Ek+2

)// · · ·

)We can pick l = −n and k ≥ 0. In this case,

πn(E) = co lim

(π0

(Ωk+nEk

) Ωk+nσk // π0

(Ωk+n+1Ek+1

) Ωk+n+1σk+1 // π0

(Ωk+n+2Ek+2

)// · · ·

)3. If n ≤ −1 then we need to pick k ≤ l ≤ −n. We can pick k = l = −n. In this case,

πn(E) = co lim(π0 (E−n)

σ−n // π0 (Ω1E−n+1)Ω1σ−n+1 // π0 (Ω2E−n+2) // · · ·

)A spectrum would be called Ω-spectrum, if σk : Ei −→ ΩEk+1 are homotopy equiv-

alence, for all k ≥ 0. In this case, all the above maps of homotopy groups/sets areisomorphisms. Therefore,

1. If n ≥ 0 then πn(E) = πn(E0).

2. If n ≤ −1, then πn(E) = π0(E−n).

4.8. SUSPENSION OF AN EXACT CATEGORY 151

4.8 Suspension of an Exact Category

Given an exact categroy E , we would first define an exact category, SE to be called thesuspension of E .

Definition 4.8.1. Let E be an exact category. The filter category F `(E ) of E is an exactcategory, defined as follows:

1. The objects of F `(E ) and sequences

A0 i0 // A1

i1 // A2 i2 // A3

// · · · ,where ir are inflesion. (4.20)

Such an object would, sometimes be denoted by A•.

2. Given another object

B• : B0 j0 // B1

j1 // B2 j2 // B3

// · · ·

in F `(E ), a morphism f• : A• −→ B• is a sequence of morphism fn : An −→ Bn

such that the obvious diagram

A0 i0 //

f0

A1 i1 //

f1

A2 i2 //

f2

A3 //

f3

· · ·

B0

j0// B1

j1// B2

j2// B3 // · · ·

commute.

3. A pair of composible morphisms:

K•f• //M•

g• // C• in F `(E )

is declared a conflasion in F `(E ) , if

Knfn //Mn

gn // Cn are conflasions in E ∀ n ≥ 0.

This defines the structure of an exact category on F `(E ).(See [K, Section 5] for more about the filtered category.

Now, we define the countable envelop of E .

Definition 4.8.2. Let E be an exact category. The countable envelop cF (E ) of E is anexact category, defined as follows:


1. The objects of cF (E ) are the same as that of the filtering category F `(E ), as inEquation 4.20.

2. Given two objects A•, B• in cF (E ), the set of morphisms is defined to be

MorcF (E )(A•, B•) := limico lim

jMorE (Ai, Bj)

One can visualize this set, as the set of limits of diagrams:

MorE (Ai, Bj)

xx &&MorE (Ai−1, Bj)

**

MorE (Ai, Bj+1)

ttMorE (Ai−1Bj+1)

Amorphism inMorcF (E )(A•, B•) would be denoted by fij where fij ∈MorE (Ai, Bj).

3. There is a natural functor Φ : F `(E ) −→ cF (E ). A pair of composable morphisms

K•fij //M•

gij // C• in cF (E )

is declared a conflasion, if it is isomorphic to the image of a conflation in F `(E ).

Further, there is a fully exact functor

ι0 : E −→ F `(E ) sending X 7→(X

1X// X

1X// X

1X// X // · · ·

)Composing, we get a fully exact functor ι : E −→ cF (E ) as follows

Eι0 //

ι""

F `(E )

cF (E )

Note, by definition of cF (E ), the vertical functor is fully exact.(A slightly different argument was given in [Sm2].)

Lemma 4.8.3. Use the notations as in (4.8.2). Consider the "inclusion" functor ι : E −→cF (E ). Then, the construction (1.4.8) applies to this inclusion, and hence the quotientcF (SE)

Eof exact categories exists.


We say that the sequence of exact functors:

E // cF (E ) // cF (SE)E

is exact.

Consequently, the sequence, of K-theory spaces

K (E ) //K (cF (E )) //K(cF (SE)

E

)is homotopy fibration.

Proof. To be inserted.

Now, we are ready to define suspension of E .

Definition 4.8.4. Let E be an exact category. Define the suspension of S(E ) of E by

S(E ) :=cF (SE)

E

It follows from (4.8.3), the sequence, of K-theory spaces

K (E ) //K (cF (E )) //K (S(E )) is homotopy fibration.

Lemma 4.8.5. Use the notations as in (4.8.2). Then, K(cF (E )) is contractible.Consequently, ΩK (S(E )) ' K (E ) is a homotopy equivalence.

Proof. (To be completed. We skip the proof that K(cF (E )) is contractible.) We have

K (E ) //K (cF (E )) //K (S(E )) is homotopy fibration.

Hence,K (E ) // ? //K (S(E )) is homotopy fibration.

Hence,ΩK (S(E )) //K (E ) // ? is homotopy fibration.

Lemma 4.8.6. Suppose E is an exact category and

Y0

iY1

jY2 be two inflations.

Then, Y1Y0−→ Y2

Y0is an inflation.

Proof. Consider the diagramY1Y0

j // Y2Y0

p

0 // Y2

Y1

Routine chasing the definitions of kernel and cokernels, it follows that j is the kernel of p.Since the bottom horizontal line is an inflation, so it the top.


4.8.1 Cofinality and Idempotent Completion

Definition 4.8.7. Let A be an additive category and B ⊆ A be a full subcategory.

1. We say that B ⊆ A is cofinal if every object in A is direct factor (i.e summand)of an object in B. We also say that B is cofinal in A .

For a commutative ring A, the category F (A) of finitely generated free modules iscofinal in P(A).

2. If A is an exact category, a fully exact subcategory B ⊆ A is said to be cofinal, ifit is cofinal in the above sense and if B is fully exact subcategory, meaning (1) B isextension closed and (2) it preserves and detects conflations.

3. An additive category A is called idempotent complete, if for all idempotent arrowsp = p2 : A −→ A there is an isomorphism ϕ : A

∼−→ X ⊕ Y such that the diagram

Ap //

ϕ o

A

ϕo

X ⊕ Y 1X 0

0 0

// X ⊕ Y commute.

4. Given an additive category A , there is a largest additive category A , such thatA ⊆ A is cofinal. In fact, A would be idempotent complete. Such an A wouldbe called the idempotent completion of A . Existence of such a completion is provedbelow (4.8.8)

Lemma 4.8.8. Let A be an additive category. Then, there is a largest additive categoryA , such that A ⊆ A is cofinal. In fact, A would be idempotent complete.

Further, if A ie exact, then A would be exact and the inclusion is cofinal, in the senseof exact categories.

Proof. Let the objects of A be the pairs (A, p) where A ∈ Obj(A ) and p = p2 : A −→ Ais an idempotent arrow in A . A morphism (A, p) −→ (B, q) is an arrow f : A −→ B inA , such that the diagram

Ap //

f

f

A

f

B q// B

commute.


The composition in A is inherited from the compositions in A .The identities in A are defined to be 1(A,p) := p.The group structure on MorA ((A, p), (B, q)) is also inherited from that in MorA (A,B).The direct product (A, p)× (B, q) = (A×B, p× q).So, A is also an additive category.Therefore, (A, p)× (B, q) = (A×B, p× q) = (A, p)⊕ (B, q).An arrow f : (A, p) −→ (A, p) in A is an idempotent if and only if f is an idempotent inA (as well fp = f = pf).

Suppose f : (A, p) −→ (A, p) is an idempotent in A . Then, the arrows

(f p− f

): (A, f)⊕ (A, p− f) −→ (A, p),

(f

p− f

): (A, p) :−→ (A, f)⊕ (A, p− f)

are inverses of each other. This follows from direct matrix multiplications:

(f p− f

)( fp− f

)= p = 1(A,p),

(f

p− f

)(f p− f

)=

(f 00 p− f

)Now, the diagram,

(A, p)f //

o

fp− f

(A, p)

o

fp− f

(A, f)⊕ (A, p− f) f 00 0

// (A, f)⊕ (A, p− f)

commutes.

the map (f 00 0

)=

(1(A,f) 0

0 0

)This establishes that A is an idempotent complete additive category.

Now, the canonical "inclusion" functor is defined by ι : A −→ A by sending X 7→(X, 1X). Given any object (X, p) ∈ Obj(A ), we can write (X, 1) ∼= (X, p)⊕ (X, 1− p). So,A ⊆ A is cofinal.

To establish that A is the largest cofinal category to A , let ζ : A −→ B be a cofinalinclusion of additive categories. For Y ∈ Obj(B), Y ⊕ X ∼= A for some X ∈ Obj(B),A ∈ Obj(A ). Define p as follows:

A //

p

Y

A

Define ζ(Y ) = (A, p)

One checks that this is well defined (always up to isomorphism).


If A is an exact category, declare a sequence

(K,w)f // (A, p)

g // (C, q) exact in A if Kf // A

g // C is exact in A

This gives a structure of an exact category on A and ι : A −→ A is fully exact (i. e.extension closed and preserves and detects conflations). The proof is complete.

We state the following theorem without proof, for now.

Theorem 4.8.9. Let A be any exact category and B ⊆ A be a cofinal fully exact subcat-egory. Then, the maps Ki(B

∼−→ Ki(A ) are isomorphism for all i ≥ 1 and injective fori = 0. In particular, this holds for the inclusion A → A .

Proof. See [Gd].

4.8.2 The K-theory Spectra

The negative K-Theory is defined as the homotopy groups of the K-theory Spectra, whichwe discuss in this section. Before that we have the following lemma.

Lemma 4.8.10. Let E be an exact category and E be its idempotent completion. Then,

the map S(E )∼−→ S(E ) is an equivalence of categories.

Consequently,

∀ n ≥ 0 Sn+1(E )∼−→ S(Sn(E ) is an equivalence of categories.

Proof. The first part follows from the construction and universal properties(1.4.8) of thequotient categories. Hence, the latter formula is valid for n = 0. Again, by n = 0 case,

Sn+1(E )∼−→ S (Sn(E ))

∼−→ S(Sn(E ) are equivalence of categories.

.

Lemma 4.8.11. Let E be an exact category, Assume E is idempotent complete. Then,

∀ n ≥ 0 ΩK(Sn+1(E )

)' K (Sn(E )) is a homotopy equivalence.

Proof. It follows inductively, from (4.8.5).

Definition 4.8.12. Let E be an exact category. Define the K-theory Spectra of E by

K(E ) :=

K(E),K(S2(E )

),K(S3(E )

),K(S4(E )

), · · ·

Also,

∀ n ∈ Z define Kn(E ) := πn(K(E ))

Appendix A

Not Used

Not Used

A.0.3 Some Jargon from [H] (SKIP)

The following is [H, Prop. 3.2.5].

Proposition A.0.13. Suppose C is a category with all small colimits. Then, the categoryC ∆ is equivalent to the category of adjunction SSet −→ C . We denote

A• ∈ C ∆ 7→ (A• ⊗−,C (A•,−), ϕ) : SSet −→ C

Proof. I will only write down the construction.

1. Given

Φ := (F,U, ϕ) : SSet −→ C consider

∆ //

A•(Φ) ""

SSet

F

C

So, A•(Φ) ∈ C ∆.

2. Given A• ∈ C ∆. We need to first, give a functor SSet −→ C .

(a) Given K ∈ SSet, ∆K ∈ SSet, where

(∆K)n = MorSSet(∆[n], K)

Also, there is a functor

∆K −→ ∆ sending and n simplex (∆[n] −→ K) 7→ [n]

There is a restriction functor

C ∆ −→ C ∆K

157

158 APPENDIX A. NOT USED

(b) There is the colimit functorC ∆K −→ C

(c) CombiningC ∆

⊗##

// C ∆K

C

For A• ∈ SSet, this diagonal associates K −→ A• ⊗K.

Corollary A.0.14. There is a pointed version of Proposition A.0.13.

159

Now we apply (A.0.13). The following is [H, Rem. 3.1.7].

Remark A.0.15. There are two remarks:

1. A simplicial analogue is possible.

2. By Proposition A.0.13, a functor C −→ C ∆ gives rise to a functor

C −→ ADJ(SSet,C )

This gives rise to a bifunctor

−⊗− : C × SSet −→ C

In deed, for A ∈ C , the functor A⊗− : SSet −→ C will have a right adjoint.

3. An important application is when C = SSet, when we consider the functor

SSet −→ SSet∆ given by A 7→ A×∆[n] ∈ SSet∆

The associate bifunctor is just

SSet× SSet −→ SSet (A,L) 7→ A× L

Its adjoint (i. e. Adjoint of (A×−) : SSet −→ SSet) is:

Map(A,L) where Map(A,L)n := MorSSet (A×∆[n], L)

4. Another example comes from Top.

(a) As usual, define

|∆[n]| :=

(t1, . . . , tn) : 0 ≤ ti ≤ 1,

n∑i=1

ti ≤ 1

=

n∑i=0

ttei : 0 ≤ ti ≤ 1,n∑i=1

ti = 1

where e0, e1, . . . , en is the standard basis of Rn+1.So, |∆[−]| ∈ Top∆ is a cosimplical topological space.

(b) By Proposition A.0.13, we get an Adjunction

(|−| , Sing, ϕ) : SSet −→ Top.

(c) Not surprisingly, |−| is called the geometric realization and Sing is called thesingular functor. While I do not have a proof, I am sure, |−| coincides with thedescription given above (Definition 3.3.2) and Sing coincides with the functorS (Definition 3.3.4).


A.0.4 On SSet Realization

Prove the following lemma, which may not be true:

Lemma A.0.16. For, σn−1 ∈ Kn−1 and σn ∈ Kn, with σn non degenerate, we have

(σn−1, x) ' (σn, y) in |K| ⇐⇒

∃ a face di ∈ ∆([n− 1], [n]), x′ ∈ |∆[n− 1]| 3 |di|(x′) = y

(σn−1, x) ' (K(di)(σn), x′) in |Kn−1|

Proof. The implication (⇐=) is obvious, because under the hypotheses (σn−1, x) ' (K(di)(σn), x′) '(σn, y). To prove the implication (=⇒), assume (σn−1, x) ' (σn, y) in |K|. It follows,

1. There are integers n0 = n − 1, n1, . . . , nk−1, nk = n, such that, for j = 1, . . . , k,nj = nj−1 ± 1.

2. Further, for j = 1, . . . , k − 1, there are (τj, xj) ∈ Knj × |∆[nj]|. We write (τ0, x0) =(σn−1, x) and (τn, y) = (σn, xk).

3. Further, for j = 1, 2, . . . , k, we have

(a) if nj = nj−1 + 1, there is a face map di : [nj−1] −→ [nj] such that

K(di)(τj) = τj−1, and |di|(xj−1) = xj

(b) if nj = nj−1 − 1, there is a degeneracy map si : [nj−1] −→ [nj] such that

K(si)(τj) = τj−1, and |si|(xj−1) = xj

A.0.5 Inclusion of Geometric Realizations

This Proof Remains Incomplete. However, we have an alternative proof, given in Proposi-tion 3.3.11 The following is a basic result on geometric realization.

Proposition A.0.17. Suppose K is a simplicial set and L ⊆ K is a simplical subset.Then, the map |L| −→ |K| is injective.

Proof. For (σ, x) ∈ Lp × |∆[p]|, (σ′, x′) ∈ Lq × |∆[q]| and (τ, y) ∈ Kn × |∆[n]|. Assume

(σ, x) ' (τ, y) ' (σ′, x′) in |K| to prove (σ, x) ' (σ′, x′) in |L|

The simplest case this problem is the following:

(σ, x) ' (τ, y) ' (σ′, x′) in |K| 3 ” ' ” are induced by a face or degeneracy.

In this case, we prove (σ, x) ' (σ′, x′) in |L|.

161

1. Suppose, one of the two (σ, x) ' (τ, y) ' (σ′, x′) equivalences, say the first one, isinduced by a map f ∈ ∆([n], [m1]). It follows,

τ = K(f)σ. Therefore, τ ∈ Ln and σ, x) ' (σ′, x′) in |L|.

2. (Recall, f ∈ ∆([m], [n]) is surjective if and only if it si product of degeneracies.)Suppose one of the two (σ, x) ' (τ, y) ' (σ′, x′) equivalences is induced by a productof degeneracies s =

∏ri=1 s

jii ,∈ ∆([m1], [n]) or s =

∏ri=1 s

jii ,∈ ∆([m2], [n]). Assume

the first one is induced by s ∈ ∆([m1], [n]). Now, s has a right inverse d. We have,

K(s)τ = σ, |s|(x) = y. =⇒ τ = K(d)K(s)τ = K(d)σ ∈ Ln

So, these relations are in |L| and hence (σ, x) ' (σ′x′) in |L|.

3. Suppose (σ, x) ' (τ, y) ' (σ′, x′) are induced by two face maps di, dj ∈ ∆([n−1], [n]).In this case, p = q = n− 1.

di, dj ∈ ∆([n− 1], [n]) be faces, and

(σ, x) ∼di (τ, y)

(σ′, x′) ∼dj (τ, y)

We have, K(di)τ = σ |di|(x) = yK(dj)τ = σ′ |dj|(x′) = y

Degeneracies in ∆([n−1], [n−2]) would be denoted by ζ i and faces in ∆([n−2], [n−1])would be denoted by δi. If i = j, then σ = σ′ and x = x′. So, we assume i < j. Wehave

K(djδi) = K(δi)K(dj) = K(δj−1)K(di) = K(diδj−1)

It follows, K(δj−1)K(di)τ = K(δj−1)(σ) |di|(x) = yK(δi)K(dj)τ = K(δi)(σ′) |dj|(x′) = y

With y = (y0, y1, . . . , yn) we have yi = yj = 0

Let x0 = |ζ isj|(y) = |ζj−1si|(y) = (y0, . . . , yi−1, yi+1, . . . , yj−1, yj+1, . . . , yn)and σ0 = K(δj−1)(σ) = K(δi)(σ

′) ∈ Ln−2

We claim(σ0, x0) = (K(δj−1)(σ), x0) ' (σ, x) given by δj−1 : [n− 2] −→ [n− 1].(σ0, x0) = (K(δi)(σ′), x0) ' (σ′, x′) given by δi : [n− 2] −→ [n− 1].

One checks,|di|(|δj−1|(x0)) = |di|(x) = y =⇒ |δj−1|(x0) = x.|dj|(|δi|(x0)) = |dj|(x′) = y =⇒ |δi|(x0) = x′.

So, the claim is established and (σ, x) ' (σ′, x′) in |L|.


4. Now suppose f ∈ ∆([m1], [n]) and g ∈ ∆([m2], [n]) be product of face maps (whichmeans injective maps) that induce these two equivalence

(σ, x) ∼f= (τ, y), (σ′, x′) ∼g= (τ, y), Write n = m1 + p = m2 + qK(f)τ = σ |f | (x) = yK(g)τ = σ′ |g| (x′) = y

f = dip · · · di2di1 0 ≤ i1 < i2 < · · · < ipg = djq · · · dj2dj1 0 ≤ j1 < j2 < · · · < jp

This completes the proof that (σn−1, x) ' (σ′n−1, x′), in the simplest case stated above.

(This Inductive Step remains incomplete. Alternately, if we can do the inductive stepin (3.3.8), then we can use that to give an alternative proof one (3.3.11), using the CWstructure (3.3.10) on |L| and |K|.)

For the general case, suppose for (σ, x) ∈ Lp × |∆[p]|, (σ′, x′) ∈ Lq × |∆[q]|, there ischain of equivalences:

(σ, x) = (τ0, y0) ∼η1 (τ1, y1) ∼η2 · · · ∼ (σr−1, yr−1) ∼η2 (τr, yr) = (σ′, x′)

where each equivalence is given by a face or degeneracy ηi, with (τi, yi) ∈ Kni × |∆[ni]|.We will prove that (σ, x) ' (σ′x′) in |L|. If each of the arrow are left to right or right toleft, then τi ∈ Lni ∀ i, and we the through.

1. Except in case (out of four) η1 : [n0] −→ [n1] is a face map, it follows τ1 ∈ Ln1 (notefor any degeneracy s K(s) has a left inverse), and induction would apply. So, assumedi11 := η1 : [n0] −→ [n1] = [n0 + 1] is a face map.

2. Now, we consider various cases of η2.

(a) Suppose η2 = sj22 : [n1] −→ [n2] is a degeneracy. Using the commutativity ofsj2di11 = di

′sj′ , we can reduce the length of the chain and we would be done.

Here τ0 = K(di11 )τ1 τ1 = K(sj22 )τ2

Bibliography

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