Introduction to Simplicial Homotopy Theory · Aran Komatsuzaki Introduction to Simplicial Homotopy Theory May 6th, 2016 10 / 32. Fibrant De nition A brant simplicial set (or Kan complex)

Introduction to Simplicial Homotopy TheoryFor The Sake of Introduction to Higher Topos Theory

Aran Komatsuzaki

May 6th, 2016

Aran Komatsuzaki Introduction to Simplicial Homotopy Theory May 6th, 2016 1 / 32

Bibliography

Simplicial Homotopy Theory, by Paul G. Goerss, John JardineHigher Topos Theory, by Jacob Lurienlab & Mathoverflow


Motivations

Why simplicial homotopy theory?

Simplicial homotopy theory is the study of homotopy theory by means ofsimplicial sets, but also the study of those properties of simplicial setsdetectable by means of techniques adapted from topological homotopytheory. It can be applied to:

Representation theory (via the stable module category)

Algebraic geometry (via motivic homotopy theory)

Graph theory (via work of Bissen and Tsemo)

Category theory (via work of Rezk, among others)

Universal algebra (via colored operads and PROPs)

Mathematical physics (via TQFTs)

Dynamical systems (via Gaucher’s work on flows)

Computer science (via work of David Spivak, among others)


Simplicial sets

Definition

Let ∆ be the category of finite ordinal numbers, with order-preservingmaps between them. Precisely speaking, the objects consist of elements n,n ≥ 0, where n is a string of relations

0→ 1→ 2→ · · · → n

A morphism θ : m→ n is an order-preserving set function, or alternativelyjust a functor. We also define simplicial set category as S := Set∆op

,and it objects are called simplicial sets.

Example

The standard n-simplex is defined as ∆n := hom∆(·,n) ∈ Obj(S). Amorphism ∆n → ∆m is induced by a morphism m→ n by Yoneda lemma.


Simplicial sets

Definition

Let ∆ be the category of finite ordinal numbers, with order-preservingmaps between them. Precisely speaking, the objects consist of elements n,n ≥ 0, where n is a string of relations

0→ 1→ 2→ · · · → n

A morphism θ : m→ n is an order-preserving set function, or alternativelyjust a functor. We also define simplicial set category as S := Set∆op

,and it objects are called simplicial sets.

Example

The standard n-simplex is defined as ∆n := hom∆(·,n) ∈ Obj(S). Amorphism ∆n → ∆m is induced by a morphism m→ n by Yoneda lemma.


Simplicial sets

Example

There is a standard covariant functor

∆→ Top

n 7→ |∆n|

where the standard n-simplex |∆n| is just

|∆n| = {(t0, · · · , tn) ∈ Rn+1 :∑

ti = 1, ti ≥ 0}

Example

If T is a topological space, the contravariant functorS(T ) : n 7→ hom(|∆n|,T ) is called singular set, and it is a familiar objectwhich gives the singular homology of the space T .


Simplicial sets

Example

There is a standard covariant functor

∆→ Top

n 7→ |∆n|

where the standard n-simplex |∆n| is just

|∆n| = {(t0, · · · , tn) ∈ Rn+1 :∑

ti = 1, ti ≥ 0}

Example

If T is a topological space, the contravariant functorS(T ) : n 7→ hom(|∆n|,T ) is called singular set, and it is a familiar objectwhich gives the singular homology of the space T .


Simplicial sets

Example

Yoneda lemma also implies that for Y ∈ Obj(S)

homS(∆n,Y ) = homS(hom∆(·,n),Y ) ' Y (n) := Yn

where Yn is the set of n-simplices of Y . Then, defined i : n− 1→ n, 0 ≤ i ≤ n and s j : n + 1→ n, 0 ≤ j ≤ n s.t.

d i (0→ 1→ · · · → n − 1) = (0→ 1→ · · · i − 1→ i + 1→ · · · → n)

s j(0→ 1→ · · · → n + 1) = (0→ 1→ · · · j → j → · · · → n)

We can easily verify cosimplicial identities (omitted). These induce ”dual”maps di : Yn → Yn−1, , 0 ≤ i ≤ n and sj : Yn → Yn+1, 0 ≤ j ≤ n withsimplicial identites.


Example

The k-th horn Λnk ⊆ ∆n (n ≥ 1) is the subcomplex of ∆n which is

generated by all faces dj(ιn) except the k-th face dk(ιn). One couldrepresent Λ2

0, for example, by the picture


Simplicial sets

Definition

For a small category C, the classifying space (or nerve) BC is thesimplicial set with BCn = homCat(n,C) which is a string

a0α1−→ a1

α2−→ · · · αn−→ an

of composable arrows of length n in C.

Example

Obj(C) = BC0, Mor(C) = BC1

φ ∈ hom(a0, a1) ⇐⇒ d0(φ) = a0 and d1(φ) = a1


Simplicial sets

Definition

For a small category C, the classifying space (or nerve) BC is thesimplicial set with BCn = homCat(n,C) which is a string

a0α1−→ a1

α2−→ · · · αn−→ an

of composable arrows of length n in C.

Example

Obj(C) = BC0, Mor(C) = BC1

φ ∈ hom(a0, a1) ⇐⇒ d0(φ) = a0 and d1(φ) = a1


Realization

Lemma

X ' lim−→∆n→X∆n (Proof: just use the fact that any functor C→ Sets on

a small category C is a colimit of representable functors)

Definition

The realization | · | : S→ Top is defined as |X | = lim−→∆n→X|∆n|

Lemma

There is a natural bijection homTop(|X |,Y ) ' homS(X ,SY ), which isnatural in X and Y .

Proof.

homTop(|X |,Y ) ' lim←−∆n→XhomTop(|∆n|,Y ) ' lim←−∆n→X

homS(∆n,S(Y ))' homS(X ,SY )


Realization

Lemma



Definition


Lemma


Proof.




Realization

Lemma



Definition


Lemma


Proof.




Realization

Lemma



Definition


Lemma


Proof.


homS(∆n, S(Y ))' homS(X , SY )


Fibrations

Example

A map p is called Kan fibration if for every commutative diagram (left),there is a dotted arrow making it commute. The map f of the middlediagram is the usual (Serre) fibration, since |Λn

k | ' Dn−1 and|∆n| ' Dn−1 × I . By adjointness (to S), the middle diagram is equivalentwith the right one (which is similar to but weaker than Serre fibration).Since |Λn

k | → |∆n| is a deformation retract, if U is a point, there exists adotted arrow in the middle diagram.

Λnk� _

i��

// X

p

��∆n

>>

// Y

|Λnk |� _

��

// T

f��

|∆n|

>>

// U

Λnk� _

��

// S(T )

S(f )

��∆n

<<

// S(U)

Lemma

Therefore, S(T )→ ∗ is a fibration.


Fibrations

Example

A map p is called Kan fibration if for every commutative diagram (left),there is a dotted arrow making it commute. The map f of the middlediagram is the usual (Serre) fibration, since |Λn

k | ' Dn−1 and|∆n| ' Dn−1 × I . By adjointness (to S), the middle diagram is equivalentwith the right one (which is similar to but weaker than Serre fibration).Since |Λn

k | → |∆n| is a deformation retract, if U is a point, there exists adotted arrow in the middle diagram.

Λnk� _

i��

// X

p

��∆n

>>

// Y

|Λnk |� _

��

// T

f��

|∆n|

>>

// U

Λnk� _

��

// S(T )

S(f )

��∆n

<<

// S(U)

Lemma

Therefore, S(T )→ ∗ is a fibration.Aran Komatsuzaki Introduction to Simplicial Homotopy Theory May 6th, 2016 9 / 32

Fibrant

Definition

A fibrant simplicial set (or Kan complex) is a simplicial set Y s.t. themap Y → ∗ is a fibration, i.e., iff every map α : Λn

k → Y can be extendedto a map defined on ∆n in the sense that there is a commutative diagram

Λnk� _

��

α// Y

∆n

>>

In particular, singular complexes are fibrant by the lemma.

Example

Define A simplicial group H to be an object of Grp∆op. Then, the

underlying simplicial set of any simplicial group is fibrant. Also, BG isfibrant if G is groupoid.


Fibrant

Definition

A fibrant simplicial set (or Kan complex) is a simplicial set Y s.t. themap Y → ∗ is a fibration, i.e., iff every map α : Λn

k → Y can be extendedto a map defined on ∆n in the sense that there is a commutative diagram

Λnk� _

��

α// Y

∆n

>>

In particular, singular complexes are fibrant by the lemma.

Example

Define A simplicial group H to be an object of Grp∆op. Then, the

underlying simplicial set of any simplicial group is fibrant. Also, BG isfibrant if G is groupoid.


Anodyne extensions

Definition

A class of monics of S is saturated if the following conditions aresatisfied: A: All isomorphisms are in M.B: M is closed under pushout.C: Each retract of an element of M is in M.D: M is closed under countable compositions and arbitrary direct sum.

Definition

A map p : X → Y is said to have the right lifting property w.r.t a classof monics M if in every diagram

A

i��

// X

p��

B

??

// Y

with i ∈ M the dotted arrow exists making it commute.


Anodyne extensions

Definition

A class of monics of S is saturated if the following conditions aresatisfied: A: All isomorphisms are in M.B: M is closed under pushout.C: Each retract of an element of M is in M.D: M is closed under countable compositions and arbitrary direct sum.

Definition

A map p : X → Y is said to have the right lifting property w.r.t a classof monics M if in every diagram

A

i��

// X

p��

B

??

// Y

with i ∈ M the dotted arrow exists making it commute.


Anodyne extensions

Lemma

The class Mp of all monics which have the left lifting property with respectto a fixed simplicial map p : X → Y is saturated.

Proof.

I will just show that the axiom B holds. Suppose we have thecommutative diagram (right). Then as below, since i ∈ Mp, there is θ,which induces the required lifting θ∗ : B ∪A C → X by the universalproperty of the pushout.

A

i��

// C

j��

// X

p

��B // B ∪A C // Y

A

i��

// X

p��

B

θ

??

// Y


Anodyne extensions

Lemma

The class Mp of all monics which have the left lifting property with respectto a fixed simplicial map p : X → Y is saturated.

Proof.

I will just show that the axiom B holds. Suppose we have thecommutative diagram (right). Then as below, since i ∈ Mp, there is θ,which induces the required lifting θ∗ : B ∪A C → X by the universalproperty of the pushout.

A

i��

// C

j��

// X

p

��B // B ∪A C // Y

A

i��

// X

p��

B

θ

??

// Y


Anodyne extensions

Definition

The saturated class MB generated by a class of monomorphisms B is theintersection of all saturated classes containing B.

Lemma

The classes Bi defined as follows have the same saturation.B1 := the set of all inclusions Λn

k ⊆ ∆n, 0 ≤ k ≤ n, n > 0.B2 := the set of all inclusions (∆n × ∂∆n) ∪ ({e} ×∆n) ⊆ (∆1 ×∆n),e = 0, 1B3 := the set of all inclusions (∆n × Y ) ∪ ({e} × X ) ⊆ (∆1 × X ) whereY ⊆ X is an inclusion of simplicial sets, e = 0, 1.

Corollary

MB1 is called the class of anodyne extensions. Therefore, a fibration is amap which has the right lifting property with respect to all anodyneextensions.


Anodyne extensions

Definition


Lemma



Corollary



Anodyne extensions

Definition


Lemma



Corollary



Function complexes

Definition

For X ,Y simplicial sets, the function complexes Hom(X ,Y ) is thesimplicial set defined by Hom(X ,Y )n := homS(X ×∆n,Y ). θ : m→ ninduces θ∗ : Hom(X ,Y )n → Hom(X ,Y )m which is defined by

(X ×∆n f−→ Y ) 7→ (X ×∆m 1×θ−−→ X ×∆n f−→ Y )

Lemma

Define an evaluation map ev : X ×Hom(X ,Y )→ Y to be s.t.(x , f ) 7→ f (x , ιn). Then, ev∗ : homS(K ,Hom(X ,Y ))→ homS(X × K ,Y )which is defined by sending g : K → Hom(X ,Y ) to

X × K1×g−−→ X ×Hom(X ,Y )

ev−→ Y

is a bijection which is natural in K ,X ,Y .


Function complexes

Definition

For X ,Y simplicial sets, the function complexes Hom(X ,Y ) is thesimplicial set defined by Hom(X ,Y )n := homS(X ×∆n,Y ). θ : m→ ninduces θ∗ : Hom(X ,Y )n → Hom(X ,Y )m which is defined by

(X ×∆n f−→ Y ) 7→ (X ×∆m 1×θ−−→ X ×∆n f−→ Y )

Lemma

Define an evaluation map ev : X ×Hom(X ,Y )→ Y to be s.t.(x , f ) 7→ f (x , ιn). Then, ev∗ : homS(K ,Hom(X ,Y ))→ homS(X × K ,Y )which is defined by sending g : K → Hom(X ,Y ) to

X × K1×g−−→ X ×Hom(X ,Y )

ev−→ Y

is a bijection which is natural in K ,X ,Y .


Simplicial homotopy

Corollary

(1) If p : X → Y is a fibration, then so is p : (K ,X )→ (K ,Y )(2) If X is fibrant then the induced map i : (L,X )→ (K ,X ) is a fibration.

Definition

Let f , g : K → X be simplicial maps. We say that there is a simplicialhomotopy from f to g if there exists h (called homotopy) s.t.

K ×∆0 = K

f

%%1×d1

��K ×∆1 h // X

K ×∆0 = K

1×d0

OO

g

99


Simplicial homotopy

Corollary

(1) If p : X → Y is a fibration, then so is p : (K ,X )→ (K ,Y )(2) If X is fibrant then the induced map i : (L,X )→ (K ,X ) is a fibration.

Definition

Let f , g : K → X be simplicial maps. We say that there is a simplicialhomotopy from f to g if there exists h (called homotopy) s.t.

K ×∆0 = K

f

%%1×d1

��K ×∆1 h // X

K ×∆0 = K

1×d0

OO

g

99


Simplicial homotopy

Lemma

If X is fibrant and K is a simplicial set, homotopy of maps K → X is anequivalence relation.


Homotopical algebras

Definition

A model category C is a category which is equipped with three classes ofmorphisms, called cofibrations, fibrations and weak equivalences whichtogether satisfy the following axioms:CM1: The category C is complete and cocomplete.CM2:

Xg //

h ��

Y

f��Z

If any two of f , g and h are weak equivalences, then so is the third.CM3: If f is a retract of g and g is a weak equivalence, fibration orcofibration, then so is f .



Definition

CM4: Suppose that we are given a commutative solid arrow diagram

U

i��

// X

p��

V

>>

// Y

where i is a cofibration and p is a fibration. Then the dotted arrow existsif either i or p is also a weak equivalence.CM5: Any map f : X → Y may be factored:(a) f = p ◦ i where p is a fibration and i is a trivial cofibration, and(b) f = q ◦ j where q is a trivial fibration and j is a cofibration.

Definition

A (co)fibration which is also weak equivalence is called trivial. X is said tobe fibrant if the map X → 1 is a fibration.



Definition

CM4: Suppose that we are given a commutative solid arrow diagram

U

i��

// X

p��

V

>>

// Y

where i is a cofibration and p is a fibration. Then the dotted arrow existsif either i or p is also a weak equivalence.CM5: Any map f : X → Y may be factored:(a) f = p ◦ i where p is a fibration and i is a trivial cofibration, and(b) f = q ◦ j where q is a trivial fibration and j is a cofibration.

Definition

A (co)fibration which is also weak equivalence is called trivial. X is said tobe fibrant if the map X → 1 is a fibration.



Definition

Cylinder object for an object A in C is a commutative triangle

A t A

i��

∇

""A σ

// A

where ∇ : A t A→ A is the canonical fold map which is the identity on Aon each summand, i is a cofibration, and σ is a weak equivalence. Then aleft homotopy of maps f , g : A→ B is a commutative diagram

A t A

i��

(f ,g)

""A

h// A



Lemma

Left homotopy of maps A→ B in C is an equivalence relation if A iscofibrant. In this case, i0 and i1 are trivial cofibrations.

Definition

A path object for an object B of a closed model category C is acommutative triangle

B

p

��B

∆//

s

<<

B × B

where s is a weak equivalence, and p is a fibration.



Example

If X is a Kan complex, then Hom(∆1,X ) is a path object for X , and Y I

is a path object for each Y ∈ Obj(CGHaus).

Definition

Maps f , g : A→ B are said to be right homotopic if there is a diagram

B

p

��A

(f ,g)//

h

<<

B × B

Corollary

Given maps f , g : A→ B, where A is cofibrant and B is fibrant, f , g areright homotopic iff they are left homotopic.



Example



Definition


B

p

��A

(f ,g)//

h

<<

B × B

Corollary




Example



Definition


B

p

��A

(f ,g)//

h

<<

B × B

Corollary




Theorem

Suppose that f : X → Y is a morphism of C s.t. X ,Y are both fibrantand cofibrant. Then f is a homotopy equivalence if f is weak equivalence.

Definition

If X ,Y are both fibrant and cofibrant, the set of homotopy classes ofmaps between such objects X and Y is denoted by π(X ,Y ). The categoryπCcf is s.t. its objects are the cofibrant and fibrants of C and ismorphisms are π(X ,Y ).

Lemma

There is a functor RQ s.t. RQX is both fibrant and cofibrant and weaklyequivalent to X , and that f 7→ RQf is well-defined up to homotopy.



Theorem


Definition


Lemma




Theorem


Definition


Lemma




Definition

Homotopy category hC is the category with the same objects as C andits morphisms are defined as homhC (X ,Y ) = π(RQX ,RQY ).


Enriched categories

Example

Let C be a category which admits finite products. Then C admits thestructure of a monoidal category where the operation ⊗ is given byCartesian product A⊗ B = A× B and the isomorphisms A,B,C are inducedfrom the evident associativity of the Cartesian product. The identity 1 isdefined to be the final object of C , and the isomorphisms.

Example

Suppose that (C ,⊗) is a right closed monoidal category. Then C isenriched over itself in a natural way if one defines MapC (X ,Y ) = Y X .

Example

Let C be the category of sets endowed with the Cartesian monoidalstructure. Then a C -enriched category is simply a category in the usualsense.


Enriched categories

Example


Example


Example



Enriched categories

Example


Example


Example



Topological category

Example

A strict 2-category is a category enriched over the cartesian monoidalcategory Cat.

Example

For X topological space and 0 ≤ n ≤ ∞, we define n-category π≤nX asfollows. The objects of π≤nX are points of X . If x , y ∈ X , then themorphisms from x to y in π≤nX are given by continuous paths [0, 1]→ Xstarting at x and ending at y . The 2-morphisms are given by homotopiesof paths, the 3-morphisms by homotopies between homotopies, and soforth. If n <∞, then two n-morphisms of π≤nX are considered to be thesame iff they are homotopic to one another.

Example

π≤0X = π0X and π≤1 = π1X (fundamental groupoid). We call π≤nX an-groupoid because every k-morphism of it has an inverse up to homotopy.



Example


Example


Example




Example


Example


Example




Definition

(∞, n)-category is ∞-category in which all k-morphisms are invertible fork > n.We will refer to (∞, 0)-categories as ∞-groupoids and (∞, 2)-categoriesas ∞-bicategories and ∞-category will refer to an (∞, 1)-category.

Example

π≤∞X are actually ∞-groupoids. The converse is accepted as a principle,which means the study of ∞-groupoids are the same as the study oftopological spaces.

Definition

A topological category is a category which is enriched over CGHaus.One of definitions of ∞-category is ∞-category ∈ Obj(CatTop).



Definition


Example


Definition




Definition


Example


Definition



∞-categories

Lemma

If K is a simplicial set, the following conditions are equivalent:(1) ∃ a small category C and an isomorphism K ' BC .(2) For each 0 < i < n we have the following:

Λnk� _

��

// X

∆n

∃!

>>

Definition

An ∞-category is a simplicial set K which has the following property: forany 0 < i < n, any map f0 : Λn

i → K admits an extension f : ∆n → K .


∞-categories

Lemma

If K is a simplicial set, the following conditions are equivalent:(1) ∃ a small category C and an isomorphism K ' BC .(2) For each 0 < i < n we have the following:

Λnk� _

��

// X

∆n

∃!

>>

Definition

An ∞-category is a simplicial set K which has the following property: forany 0 < i < n, any map f0 : Λn

i → K admits an extension f : ∆n → K .


∞-categories

Example

Any Kan complex is an ∞-category. In particular, if X is a topologicalspace, then SX is ∞-category. Since BC is ∞-category, ordinary categoryis just a special type of ∞-category.

Definition

A functor F : C→ D between topological categories is a strongequivalence if it is an equivalence in the sense of enriched categorytheory. In other words, F is essentially surjective and inducehomeomorphisms MapC (X ,Y )→ MapD(F (X ),F (Y )) ∀X ,Y ∈ C .

Definition

Let C be the category with CW complexes as objects, where MapC(X ,Y )is the set of continuous maps with compact-open topology. We callH := hC the homotopy category of spaces.


∞-categories

Example


Definition


Definition



∞-categories

Example


Definition


Definition



Simplicial categories

Definition

A simplicial category is a category enriched over S. The category ofsimplicial categories will be denoted by Cat∆

Definition


Definition

The (enriched) homotopy category of C is H-enriched category whoseobjects are of C s.t. for X ,Y ∈ C

MaphC(X ,Y ) = [MapC(X ,Y )]

and the composition law is determined by θ : CGHaus→ H.



Definition


Definition


Definition






Definition


Definition


Definition






Definition

S-enriched category C is called simplicial model category if theunderlying category C0 has a structure of model category such that forevery cofibration i : A→ B and every fibration p : X → Y in C0, themorphism

C (B,X )i∗×p∗−−−→ C (A,X )×C(A,Y ) C (B,Y )

is a fibration, and such that this fibration is an acyclic fibration whenevereither i or p are acyclic.

Definition

f : X → Y is weak equivalence of simplicial sets if the induced map|X | → |Y | is weak equivalence of topological spaces.



Definition

S-enriched category C is called simplicial model category if theunderlying category C0 has a structure of model category such that forevery cofibration i : A→ B and every fibration p : X → Y in C0, themorphism

C (B,X )i∗×p∗−−−→ C (A,X )×C(A,Y ) C (B,Y )

is a fibration, and such that this fibration is an acyclic fibration whenevereither i or p are acyclic.

Definition

f : X → Y is weak equivalence of simplicial sets if the induced map|X | → |Y | is weak equivalence of topological spaces.


Important results

Theorem (Quillen)

The unit and counit morphisms X → S |X | and |SY | → Y are weakequivalence for X ∈ S and Y ∈ CGHaus.

Theorem

The following have natural model structures:

S

CGHaus

Top (Hurewicz fibrations)

Categories of presheaves of simplicial sets

Categories of quasi-categories

Furthermore, the first two are simplicial model category.


Important results

Theorem (Quillen)

The unit and counit morphisms X → S |X | and |SY | → Y are weakequivalence for X ∈ S and Y ∈ CGHaus.

Theorem

The following have natural model structures:

S

CGHaus

Top (Hurewicz fibrations)

Categories of presheaves of simplicial sets

Categories of quasi-categories

Furthermore, the first two are simplicial model category.


Important results

Theorem (Dold-Kan)

sAb ' Ch+

Homotopical algebra is just an extension of homological algebra tohomotopy theory!

remark

Model category is a context for doing homotopy theory. Along withclassical homotopy theory (on CGHaus), one can perform homotopytheory on other categories as listed in the above theorem.

Lemma

If f : X → Y is a morphism of a simplicial model category C whichinduces an isomorphism in the homotopy category hC, then f is a weakequivalence.


Important results

Theorem (Dold-Kan)

sAb ' Ch+


remark


Lemma



Important results

Theorem (Dold-Kan)

sAb ' Ch+


remark


Lemma



Documents

Introduction to Simplicial Homotopy Theory · Aran Komatsuzaki Introduction to Simplicial Homotopy Theory May 6th, 2016 10 / 32. Fibrant De nition A brant simplicial set (or Kan complex)