Download pdf - Notes on Categories - mimosa.pntic.mec.esmimosa.pntic.mec.es/jgomez53/matema/docums/villarroel-categories.pdf1.1. Deﬁnition and Examples 5 5. The category Ring of rings with unit,

Notes on Categories

Rafael Villarroel [email protected]

18th August 2004

ii

Contents

I Basic Definitions 1

1 Categories 31.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Small Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Special Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Special Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Functors 152.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Contravariant Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Isomorphism of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Types of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Natural Transformations 233.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 The Functor Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Equivalence of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

II Limits 31

4 Limits and Colimits 334.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Limit and Colimit as Functors . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 Preservation of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

iii

iv Contents

5 Universals and Adjoints 435.1 Universals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Adjoint Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 More on Limits 496.1 Limits in a Functor Category . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.3 Ends in a Functor Category . . . . . . . . . . . . . . . . . . . . . . . . . . 566.4 Iterated Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.5 Coends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

III Extras 65

7 Abelian Categories 677.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2 Chain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.3 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.4 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.5 Split Chain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.6 Exact Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8 Appendix: Calculations 758.1 The Grothendieck Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Bibliography 77

Part I

Basic Definitions

1

1Categories

1.1 Definition and Examples

In this section we provide the definition of category and many examples of categorieswhich may already be familiar to the reader.

1.1 Definition. A category C consists of

1. A class obj C, whose elements are called objects.

2. A set homC(A,B) for every pair of objects A,B, whose elements are called mor-phisms, or maps from A to B. We can call them also C-maps.

3. For every triad of objects A,B,C, a function (called composition)

homC(A,B)× homC(B,C) → homC(A,C) (1.1)

whose value at (f,g) will be denoted by g f.

4. For every object A, a distinguished element 1CA ∈ homC(A,A), called identity on

A.

They have to satisfy the following conditions:

(a) For every pair of objectsA,B and for every f ∈ homC(A,B), we must have f1CA =

1CB f = f.

(b) For every A,B,C,D ∈ obj D and f ∈ homC(A,B), g ∈ homC(B,C) and h ∈homC(C,D) we have that h (g f) = (h g) f.

3

4 Chapter 1. Categories

An identity 1CA is usually denoted by 1A if the category C is clear from the context. If

A and B are objects in the category C, then f : A → B will mean that f is an element ofhomC(A,B). In that case, we say that A is the domain of f and that B is the codomain off, and we write A = dom f, B = cod f.

An element of obj C will be called a C-object, similarly a morphism in C will be calleda C-morphism.

We note that we always consider different hom sets to be disjoint. That is, if f,g aremorphisms in a category C such that f = g, with f ∈ homC(A,B), g ∈ homC(C,D), thenwe must have that A = C, B = D.

Our first example is the prototype of all categories.

Example 1.1The category Set of sets, such that obj Set is the class of all sets and homSet(A,B) is the set of allmaps from A to B. Composition is the usual composition of maps, and for a set X, the identity1X is the identity map. Then clearly the conditions (a) and (b) of the definition of categories aresatisfied.

In the following examples of categories, the objects are sets with some extra struc-ture, and morphisms are maps which preserve that structure. Those categories are calledconcrete, in them, the compositions are given by the usual composition of maps, and theidentities are the identity maps. Since the usual composition is associative and the identitymap is a neutral element under composition, in order to prove that we have a categoryone only has to check that the composition of two morphisms is a morphism, and that theidentity map is a morphism.

Examples from Algebra

Examples 1.21. The category Grp of all groups, where we have that obj Grp is the class of all groups and

homGrp(A,B) is the set of group homomorphisms from A to B.

2. Fixing a group G, we have the category GSet, where obj GSet is the class of (left) G-setsand homGSet(X, Y) is the set of equivariant maps.

3. The category Ab of all abelian groups, such that obj Ab is the class of all abelian groupsand homAb(A,B) is the set of group homomorphisms from A to B.

4. The category Rng of rings, such that obj Rng is the class of all rings, homRng(A,B) is theset of ring homomorphisms from A to B.

1.1. Definition and Examples 5

5. The category Ring of rings with unit, such that obj Ring is the class of all rings with unitand homRing(A,B) is the set of all ring homomorphisms from A to B such that f(1) = 1.

6. The category R-mod, where R is a ring, such that objR-mod is the class of all left modulesover R and homR-mod(A,B) is the set of R-linear morphisms from A to B. We similarlyhave a category mod-R of right R-modules. Note that if R is in fact a field, then R-mod isthe category of vector spaces over R, and the morphisms in this particular case are lineartransformations.

7. The category R-alg of R-algebras, such that objR-alg is the class of all algebras over R andhomR-alg(A,B) is the set of morphisms of R-algebras from A to B.

Examples from Topology

Examples 1.31. The category Top of topological spaces, such that obj Top is the class of all topological

spaces and homTop(X, Y) is the set of continuous maps from X to Y.

2. The category Top∗ of pointed topological spaces, such that obj Top∗ is the class ofall pairs (X, x0), where X is a topological space and x0 is a point in X. We puthomTop∗

((X, x0), (Y,y0)

)as the set of continuous maps from X to Y such that f(x0) = y0.

3. The category Haus of Hausdorff topological spaces, such that obj Haus is the class of allHausdorff topological spaces and homHaus(X, Y) is the set of all continuous maps from X

to Y.

4. The category Metric of metric spaces, such that obj Metric is the class of all metric spacesand homMetric(X, Y) is the set of all continuous maps (satisfying an ε-δ definition) from X toY.

Examples from Combinatorics

Examples 1.41. The category Poset, with class of objects the class of partially ordered sets and

homPoset(P,Q) is the set of monotone maps from P to Q.

2. The category Graph where obj Graph is the class of all graphs, and for G1,G2 ∈ obj Graph,we have that homGraph(G1,G2) is the set of functions from G1 to G2 that preserve adjacency.

3. The category DirGraph where obj DirGraph is the class of all directed graphs, and forG1,G2 ∈ obj Graph, we have that homGraph(G1,G2) is the set of functions f such thatwhenever a→ b is and edge in G1, then f(a) → f(b) is an edge in G2.

4. The category SimplComplex, with class of objects the class of abstract simplicial complexesand homSimplComplex(K,L) is the set of simplicial maps from K to L.


At this point, we should now provide examples of categories which are not concrete. Inthe following examples, either the objects are not sets with structure or the hom sets arenot composed by maps between sets.

Examples 1.51. We define a category mod where obj mod consists of pairs (R,M) where R is a ring and M

is a left R-module. A morphism (R,M) → (S,N) in mod is a pair of maps (φ, f), whereφ : R → S is a morphism of rings and f : M → N is an additive map such that f(rm) =

φ(r)f(m) for all r ∈ R, m ∈M. We define composition as (φ ′, f ′) (φ, f) = (φ ′ φ, f f ′),which can be checked it is well-defined, and 1(R,M) = (1R, 1M).

2. Consider the category Toph, where the class of objects is the class of topological spaces andthe set homToph(X, Y) is the set of homotopy classes of continuous maps from X to Y. Thecomposition of homotopy classes is defined to be the class of the composition of arbitraryrepresentatives. This is well defined, since homotopy of maps is an equivalence relationpreserved by composition. See [ML98, page 52]. We set 1Toph

X as the homotopy class of theidentity map X→ X. The properties of homotopy let us check the conditions for a category.

3. Let Rel be a category such that obj Rel = obj Set, and for sets X, Y, we have thathomRel(X, Y) is the set of relations from X to Y. If R : X→ Y and S : Y → Z, then SR : X→ Z

is given by

S R = (x, z) ∈ X× Z | there is y ∈ Y with (x,y) ∈ R, (y, z) ∈ S . (1.2)

We put 1X = (x, x) | x ∈ X . It is straightforward to check that we have just defined acategory.

We will have plenty of more examples in the next section.

Exercises 1.1

1. Let C be a category, and A ∈ obj C. If h ∈ homC(A,A) has the property that h f = f for everyf ∈ homC(A,A), then h = 1C

A.

1.2 Small Categories

1.2 Definition. A category C is small if obj C is a set.

1.2. Small Categories 7

All the categories defined in the previous section are not small, hence they are called large.

Consider the following examples of small categories:

Examples 1.61. The empty category 0, with no objects (hence, no morphisms).

2. The category 1 such that obj 1 = ∗ (a set with one element), and hom1(∗, ∗) = 1∗. Thecomposition is then uniquely defined. The composition is uniquely defined if it is going tosatisfy (a) from Definition 1.1.

3. The category 2 with obj 2 = ∗, ∗ ′ and one non-identity map ∗ → ∗ ′. Again, the composi-tion can only be defined in one way in order form 2 to be a category.

4. Similarly, we can define a small category by means of the diagrams•

• // •

,• //

•

•, or

•//

// • . In every case, each dot represents a different object in the category being defined,and the arrows are the only non-trivial morphisms.

5. Let G be a group. We define a category G by obj G = ∗, homG(∗, ∗) = G, compositionequal to the group multiplication and 1∗ = identity element of G. We say that G is thecategory associated to the group G.

6. Let P be a preordered set, that is, a set with a reflexive and transitive relation denoted by ≤.We define a category P by putting obj P = P, and

homP(x,y) =

(x ≤ y) if x ≤ y∅ if x 6≤ y

The composition will be given by (y ≤ z) (x ≤ y) = (x ≤ z), and 1x = (x ≤ x). We saythat P is the category associated to the preordered set P.In particular, if n ≥ 0, we will use [n] to denote the category associated to the subposet0, 1, . . . ,n of N ∪ 0 with the usual order relation.

7. Let X be a topological space, then we can define a category X with obj X = X, andhomX(x,y) the set of homotopy classes of continuous maps (paths) γ : [0, 1] → X withγ(0) = x, γ1 = y. The composition is defined in representatives as the usual concatenationof paths and the identity 1x is the class of the constant path with value x.

1.3 Definition. A discrete category is a category in which all the morphisms are identities.

There can be discrete large categories, but usually this definition will be of use to uswhen we consider any set X as a small discrete category.


1.3 Constructions

We show now some ways of forming new categories from old ones.

1.4 Definition. Let C be a category and A ∈ obj C. We define the comma category A ↓ C, whereobjA↓C is the class of all morphisms in C with domain A. If f : A→ B, f ′ : A→ B ′ ∈objA↓C, then homA↓C(f, f ′) is the set of C-morphisms φ : B→ B ′ such that φf = f ′.Using diagrams, this last statement is equivalent to saying that the following diagram:

B

φ

A

f88pppppp

f′ &&NNNNNN

B ′

(1.3)

commutes.

It is entirely possible that the same C-morphism φ makes commute two different dia-grams of the form 1.3. However, as noted after Definition 1.1, we must consider them astwo different A↓C-morphisms.

Example 1.7As an example of a comma category, let ∗ be a topological space with just one point. Then∗ ↓ Top consists of maps of the form ∗ → X, which can be identified with X together with achoice of a basepoint (the image of ∗). And the morphisms in ∗↓Top are continuous maps thatpreserve the basepoint. Hence, in some sense, ∗↓Top can be identified with the category Top∗.

1.5 Definition. Similarly, if C be a category and A ∈ obj C, we define a comma category C↓A, whereobj C↓A is the class of all morphisms in C with codomain A. If h : B → A,h ′ : B ′ →A ∈ obj C ↓A, then homC↓A(h,h ′) is the set of C-morphisms ψ : B → B ′ that makethe following diagram commute:

B

ψ

h

&&NNNNNN

A

B ′h′

88pppppp

(1.4)

1.3. Constructions 9

1.6 Definition. Let C be a category. The opposite category Cop has as objects the same class of objectsas C, the hom sets are defined by homCop(A,B) = homC(B,A), the composition g fin Cop is defined to be equal to the composition f g in C, and the identities are thesame as in C.

Example 1.8

It is straightforward that( •

• // •

)op=

• //

•

•.

1.7 Definition. Let C and D be categories. The product category C × D has as objects the pairs(A,A ′) with A ∈ obj C, A ′ ∈ obj D, the morphisms from (A,A ′) to (B,B ′) are pairsof morphisms (f, f ′) with f : A → B in C and f ′ : A ′ → B ′, composition is definedcomponentwise, and 1(A,A′) = (1A, 1A′).

Example 1.9For example, if G1, G2 are groups, the category associated to the direct product of groupsG1 ×G2 is the product category G1 ×G2

More generally, for any set of n-categories C1, . . . , Cn, we can define a category C1 ×· · · × Cn. We will denote the product C× · · · × C with n factors as C×n.

1.8 Definition. Let C be a category such that there is an equivalence relation ' defined on eachhomC(A,B) for any pair of objects A, B in C, with the property that if f, f ′ ∈homC(A,B) with f ' f ′ and g : A ′ → A, h : B → B ′ are maps, then hfg ' hf ′g.(In this case ' is called a congruence in C). Then there is a quotient category C/'with the objects the same objects as in C, the hom sets are the equivalence classes un-der the relation ', the composition is defined by composition of representatives, andfor any A ∈ obj(C/') = obj C, the identity on A is the class of 1A in homC(A,A).

Example 1.5.2 is an instance of this definition.


1.4 Subcategories

1.9 Definition. Let C, C ′ be categories. We say that C ′ is a subcategory of C if obj C ′ is a subclass ofobj C, homC′(A,B) is a subset of homC(A,B) for all A, B in obj C ′, the compositionsin C ′ are defined and are restrictions of the corresponding compositions in C, and theidentity morphisms in C ′ are the ones that are so in C.If C ′ is a subcategory of C and we have that homC′(A,B) = homC(A,B) for all A, Bin obj C, we say that C ′ is a full subcategory.

As a trivial example, note that any category is a (full) subcategory of itself.

Examples 1.101. The category Set is a subcategory of the category Rel, since every map of sets is in particular

a relation. But it is not a full subcategory, since it is clear that not every relation is a setmap.

2. The category Ab is a full subcategory of Grp, since every morphism of groups betweenabelian groups is a morphism of abelian groups.

3. The category Ring is a subcategory of Rng but not a full subcategory, since for two ringswith unity R, S, the constant map R→ S sending all R to the zero element in S is a morphismin Rng but not in Ring.

4. The category Haus is a full subcategory of the category Top.

5. Neither of the categories Grp, Ring is a subcategory of the other.

Given a category C, a full subcategory is completely determined by its class of objects.In this way, we can easily define, for example, the category FinSet, which is the full sub-category of Set where the class of objects is the class of all finite sets. For any concretecategory C, we define then a category FinC.

And we can speak of the categories of finitely generated groups, of torsion abeliangroups, of noetherian rings, of finite-dimensional vector spaces, of compact Hausdorfftopological spaces, and so on. In each case, we first set an ambient category, and then wespecify the objects of the full subcategory.

1.10 Definition. We will denote with ∆ the full subcategory of Poset such that obj ∆ = [n] | n ∈ N ∪ 0 .

1.5. Special Objects 11

1.5 Special Objects

We now start studying specific properties that an object in a fixed category could have ornot have.

1.11 Definition. Let C be a category. We say that A ∈ obj C is an initial object if homC(A,C) hasexactly one element for each C ∈ obj C. We say that B ∈ obj C is a final object ifhomC(C,B) has exactly one element for each C ∈ obj C. Finally, an object which isboth an initial and a final object is called a zero object.

Examples 1.111. In Set, the empty set is the only initial object, and the final objects are exactly the sets with

only one element. A similar situation happens in Top. In particular, there are no zeroobjects either in Set or Top.

2. In Grp, the trivial group is a zero object.

3. If P is a category coming from a preordered set P as in example 6 of 1.6, an initial object inP corresponds to a minimum element of P and a final object in P with a maximum elementof P.

1.12 Definition. Let C be a category with a zero object. Then in any homC(A,B) there is a well-definedzero morphism, 0 : A → B which is the composition A → 0 → B. It can be shownthat the zero morphism is independent of the choice of the zero object.

Example 1.12If G1,G2 ∈ obj Grp, the zero morphism in homGrp(G1,G2) is the homomorphism that sendsevery element of G1 to the identity.

Exercises 1.2

1. Let C be a category with a zero object 0. If 0 ′ is another zero object, the compositions A →0 → B and A→ 0 ′ → B are equal.

2. Give an example of a category that has neither an initial object nor a final object.


1.6 Special Morphisms

And now we start studying specific properties that morphisms in a fixed category couldhave or not have.

1.13 Definition. Let f : A→ B be a morphism in a category C. We say that f is an isomorphism if thereis a C-morphism h : B → A such that h f = 1A and f h = 1B. In this case, we saythat A and B are isomorphic, and we write A ∼= B.

For example, in the category Set, a morphism f : A → B is an isomorphism exactlywhen it is bijective. Hence in a concrete category, a necessary condition for a morphismto be an isomorphism is that it is bijective.

In our algebraic examples of categories (Examples 1.2), the isomorphisms are exactlythe bijective morphisms.

In Top and Haus isomorphisms are called homeomorphisms. It is well known, how-ever, that it is possible to have a continuous and bijective map f : X → Y without X andY be homeomorphic. A morphism f : (X, x0) → (Y,y0) in Top∗ is an isomorphism if andonly if f : X→ Y is a homeomorphism. An isomorphism in Metric is called an isometry.

Finally, note that in our examples from combinatorics (examples 1.4), in general weneed more that a bijective morphism to have an isomorphism.

1.14 Definition. Let f : A→ B be a morphism in a category C. Then

1. f is monic if fg1 = fg2 implies g1 = g2, for any object C and any g1,g2 : C→ A.

2. f is epic if h1 f = h2 f implies h1 = h2, for any object D and any h1,h2 : B→ D.

It is easy to see that in a concrete category C, an injective map f : A→ B is necessarilymonic. For if g1,g2 : C → A are two C-morphisms, then g1(c) 6= g2(c) for some c ∈ Cimplies that f

(g1(c)

)6= f

(g2(c)

). Similarly, surjective maps are epic.

1.15 Definition. Let C be a category, andA an object in C. We define a preorder (reflexive and transitiverelation) in the class of monics with codomain A by declaring f ≤ g if there is a k suchthat f = g k. We write f ∼ g if f ≤ g and g ≤ f. (It then follows that k is anisomorphism). The relation ∼ is an equivalence relation and its equivalence classes arecalled the subobjects of A. Similarly, define now a preorder in the class of epics havingdomain A by f ≥ g if there is a k such that f = k g, and f ∼ g if f ≥ g and g ≥ f.The equivalence classes of ∼ are then called the quotient objects of A.

1.6. Special Morphisms 13

Exercises 1.3

1. Isomorphism is an equivalence relation on the class obj C.

2. If f : A→ B is an isomorphism in a category C, the map h in Definition 1.13 is unique.

3. Any two initial objects in a category C are isomorphic.


2Functors


2.1 Definition. Let C and D be categories. A functor F from C to D, denoted F : C → D, is composedof

1. A map (which we denote by F) from obj C to obj D,

2. For every A,B ∈ obj C, a map (also denoted by F) from the set homC(A,B) to theset homD(FA, FB)

that satisfy

(a) F(gf) = F(g)F(f) for every pair of morphisms f,g ∈ C such that gf is defined,

(b) F(1A) = 1FA for every A ∈ obj C.

So, a functor is a map on objects which preserves compositions and identities. Note thatwe sometimes omit the parenthesis, and write FA for the image of the object A ∈ objD,and similarly for the images of the morphisms.

For our examples, let us begin with general constructions.

Examples 2.1

15

16 Chapter 2. Functors

1. For every category C, we have the identity functor 1C : C → C, such that 1CA = A forevery A ∈ obj C, and 1Cf = f for every morphism in C.

2. If C ′ is a subcategory of C, there is an inclusion functor C ′ → C, which sends every objectand every morphism to itself.

3. For any two categories C, D and object B in D, we have a constant functor FB : C → Dgiven by FB(A) = B, FB(f) = 1B, for all objects A and morphisms f in C.

4. If C and D are categories, there is a projection functor pC : C × D → C, sending an object(A,A ′) of the product to A, and a morphism (f, f ′) to f.

5. For any category C, we have the diagonal functor ∆ : C → C×C defined by ∆(C) = (C,C)

and ∆(f) = (f, f).

6. For any comma categoryA↓C, there is a functor F : A↓C → C, sending the object f : A→ B

to B, and a morphism in A↓C to itself. And similarly, there is a functor from C↓A→ C,

7. If ' is a congruence in C (see Definition 1.8), there is a quotient functor C → C/', sendingeach object to itself and each morphism to its equivalence class.

The next examples are so important that deserve a box by themselves.

Example 2.2If C is a concrete category, there is a functor C → Set, sending an object A in C, which is a setwith structure, to its underlying set. A morphism in C is sent to the map between the underlyingsets. In other words, we just forget about the structure in objects of C, hence this functor is aforgetful functor.Similarly, we have a forgetful functor Rng → Ab, given by retaining the additive structure in aring and forgetting the product, and forgetful functors R-alg → Rng, R-alg → R-mod.

Example 2.3Let C be any category, and A ∈ C a fixed object. Then there is a functor C → Set, denoted ashomC(A, −), defined on objects by B 7→ homC(A,B), and on a C-morphism like f : B→ B ′, byf∗ : h 7→ f h.Sometimes the hom sets have extra structure, for example, if A and B are abelian groups, thenhomAb(A,B) has an structure of an abelian group and the map f∗ is an abelian group map.Hence in this case we get a functor homAb(A, −): Ab → Ab.We could even go further in another direction: for any category C, there is a functorhomC : Cop × C → Set, sending (A,B) → homC(A,B) and sending the pair (φ,ψ), that is amorphism from (A,B) to (A ′,B ′), to the set map homC(φ,ψ) : homC(A,B) → homC(A ′,B ′)with correspondece rule h 7→ ψhφ (remember that φ, being in Cop, goes from A ′ to A whenconsidered in C.)

We now define particular functors, classified by area.


Functors in Algebra

Examples 2.41. We define a functor F : Grp → Ab. For a group G, consider G ′ be its commutator subgroup,

and the quotient G/G ′, which is an abelian group, so we put FG = G/G ′. If f : G → H

is a group homomorphism, then f(G ′) is a subgroup of H ′, so a map f : G/G ′ → H/H ′ isinduced. We define F(f) = f, it is then straightforward to check it is indeed a functor.

2. The functor U : Ring → Grp such that U(R) = the group of invertible elements in R, and iff : R→ S is a ring homomorphism, then U(f) : U(R) → U(S) is defined as U(f) = f|U(R).

3. Let ρ : R→ S be a morphism of rings. Then we have a functor ρ# : S-mod → R-mod definedthe following way: Let N be an S-module. Then ρ#N has as abelian group the same N, andthe action of r is: rn = ρ(r)n, where in the right side we have the action of S. Furthermore,ρ#(f) = f.

4. Fix a ring R. Then we have a functor F : Grp → R-alg, that sends a group to its groupalgebra with coefficients in R.

Functors in Topology

Examples 2.51. There is a functor Metric → Haus, sending each metric space (X,d) to the associated

topological space, which is always Hausdorff.

2. We have the so called loop functor Ω : Top∗ → Top∗, whose value at (X, x0) is the pair(Ω(X, x0), cx0

), where Ω(X, x0) = f : [0, 1] → X | f(0) = f(1) = x0 , with the compact-

open topology, and cx0 is the constant map with value x0. If φ : (X, x0) → (Y,y0) is a mapin Top∗, then Ω(φ)(f) = φ f.

3. There is the reduced suspension functor Σ : Top∗ → Top∗, whose value at (X, x0) is com-posed by the topological space given by the quotient (X×[0, 1])/(X×0∪X×1∪x0×[0, 1])

with the base point being the corresponding to the collapsed subspace.

Functors in Combinatorics

Examples 2.61. There is a functor K : Poset → SimplComplex, defined on P as the complex with simplices

the totally ordered subsets of P P. An order preserving map f : P → P ′ induces a simplicialmap K(f) : K(P) → K(P ′), given by K(f)(x0, . . . , xn) = f(x0), . . . , f(xn).


2. We have a functor ∆ : Graph → SimplComplex, such that for a graph G, we define ∆(G)

as the simplicial complex with simplices the vertices of complete subgraphs of G. Since anymap in Graph, say f : G → G ′, sends complete subgraphs to complete subgraphs, we havea well defined simplicial map ∆(f) : ∆(G) → ∆(G).

3. We have a functor Graph → Graph sending each graph G to its dual graph G∗. This workssince a graph map f : G→ G ′ induces a map in the duals f∗ : G∗ → (G ′)∗.

But by far the most interesting examples are the functors that cross the boundaries of anarea of study.

Examples 2.71. We have a functor F : Grp → Poset, that sends a group G to the set of subgroups of G,

partially ordered by containment. A morphism of groups f : G → H sends subgroups tosubgroups preserving containment, so F(f) is defined.

2. There is a functor π1 : Top∗ → Grp, that sends a pointed topological space to its fundamen-tal group, and a pointed continuous maps to a homomorphism between the correspondinggroups.

3. Fixing a ring R, for each n ∈ N∪ 0 there is a functor Hn(−,R) : Top → R-mod, called n-thhomology.

4. There is a functor SimplComplex → Top, called the geometric realization functor.

Next, let us consider examples that involve small categories.

Examples 2.81. If C is a discrete category, a functor F : C → D is a collection of D-objects, indexed by the

objects of C.

2. If C is the category• //

•

•, a functor F : C → D can be identified with a diagram in D of

the form

Af//

g

B

C

(2.1)

similarly, if C is now the category•

• // •

or •//

// • , then a functor F : C → D can be

identified withA

f

Bg// C

or A

f//

g// B (2.2)


respectively

3. Let G1, G2 be groups, and G1 G2 the corresponding associated categories. Then a functorF : G1 → G2 corresponds to a homomorphism G1 → G2.

4. Let G is a group and G be its associated category. Then a functor F : G → Set correspondsto a choice of a set X (the value of F(∗)), together with a collection of maps g = F(• g−→•) : X→ Xg∈G such that the group operation corresponds to composition, and the neutralelement to the identity map. This is exactly the same as having an structure of G-set on X.Hence, for any category C, it seems natural to define a G-object as a functor F : G → C.

Having defined the concept of functors, we define some more categories.

Examples 2.91. Functors can be composed, this is, if we have categories C1, C2 and C3, and functors

F1 : C1 → C2 and F2 : C2 → C3, then there is a functor F2 F1 : C1 → C3, defined asthe composition of the corresponding maps on objects and on morphisms. Moreover, thiscomposition is associative. This, together with the identity functor for any category C allowus to define the category SCat, with objects the small categories and the set of morphismsbetween two categories C and D is the collection of functors between C and D. Since afunctor C → D is determined by a map obj C → obj D and both obj C, obj D are sets, sois homSCat(C, D). Hence the restriction to small categories is in order to have that the homsets are, precisely, sets.

2. Let F : C → D be a functor, and A a fixed object of D. We define the category A ↓ F, withobjects the pairs (B, f) | B ∈ obj C, f ∈ homD(A, FB) . A morphism from (B, f) to (B ′, f ′)is a C-morphism φ : B→ B ′ such that the diagram

FB

F(φ)

A

f 77oooooo

f′ ''OOOOOO

FB ′

(2.3)

commutes. In other words, A↓C is just A↓1C. We similarly could define a category F↓A.

And then we can define more functors.

2.2 Definition. Given a functor F : C → D, and A a fixed object of D, we have a projection functorsF ↓ A → C, defined on objects sending (B, f) to B, and on morphisms by sendingφ : (B, f) → (B ′, f ′) to φ.

We similarly have a projection functor F↓A→ C.


Exercises 2.1

1. Give an example of a category C and a functor F : C → C such that F(C) = C for all C ∈ obj C,but F 6= 1C.

2. If we have functors F : C → D and F ′ : C ′ → D ′, then we can define a functor (F, F ′) : C× C ′ →D×D ′, which in objects is (F, F ′)(C) =

(F(C), F ′(C)

).

2.2 Contravariant Functors

The reader would have probably noted that if we fix the second variable in the definitionof the hom functor (Example 2.3), we get a functor Cop → Set. This situation deserves adefinition.

2.3 Definition. Let C an D be categories. A contravariant functor F : C → D is a functor Cop → D.That is, F consists of a map from obj C to obj D, and for every A,B ∈ obj C, mapsfrom homC(A,B) to homD(FB, FA) such that F(gf) = F(f)F(g) for every morphismsf,g ∈ C such that g f is defined, and F(1A) = 1FA for every A ∈ obj C.

The most important examples of a contravariant functor is the following: Let C be acategory, and choose B ∈ obj C. We get a contravariant functor hom(−,B) : C → Set, suchthat hom(−,B)(A) = hom(A,B), and if f : A→ C, then

hom(−,B)f : hom(C,B) → hom(A,B) (2.4)

is given by g 7→ g f. We will denote hom(−,B)f by f∗.

Example 2.10We single out one particular case of the previous note. If R is a ring, for any left R-module Mone has that homR-mod(M,R) = M∗ has a natural structure of a right R-module, furthermore, ifφ : M → N is a morphism in R-mod, then φ∗ : N∗ → M∗ is a morphism in mod-R. Hence weobtain a contravariant functor R-mod → mod-R. On the other hand, if M is a right R-module,then hommod-R(M,R) has a natural structure of left module, and hence in this case we get acontravariant functor mod-R→ R-mod.

2.3. Isomorphism of Categories 21

In this context, the functors defined in section 2.1 are called covariant functors.

Note that the composition of two contravariant functors is a covariant functor. Inparticular, composition of the contravariant functors described in Example 2.10 results ina (covariant) functor R-mod → R-mod.

2.3 Isomorphism of Categories

2.4 Definition. Let C and D be categories. We say that C is isomorphic to D, if there are functorsF : C → D and F ′ : D → C such that F ′ F = 1C and F F ′ = 1D. We write then C ∼= D.

Examples 2.111. The categories Z-mod and Ab are isomorphic.

2. If R is commutative, R-mod and mod-R are isomorphic.

3. The categories ∗↓Top and Top∗ are isomorphic.

We can then see that a category is concrete if and only if it is isomorphic to a subcategoryof Set.

Exercises 2.2

1. For any category C, (Cop)op ∼= C.

2. For any categories C and D, C×D ∼= D× C.

3. For any categories C and D, (C×D)op ∼= Cop ×Dop.

4. For any categories C, D and E, (C×D)× E ∼= C× (D× E)

5. If C and D are categories and F : C → D is a functor, show that (A↓F)op ∼= F↓A.

6. If C and D are categories and F : C → D is a functor, show that if f is an isomorphism in C, thenF(f) is an isomorphism in D. If A is an initial object in C, is F(A) an initial object in D?

7. Give an example of a category C such that C is isomorphic to Cop, and another example whenthey are not isomorphic.


2.4 Types of Functors

2.5 Definition. Let F : C → D be a functor. We say that F is

1. faithful, if for each A,B ∈ obj C, the map F : homC(A,B) → homD(FA, FB) isinjective.

2. full, if for each A,B ∈ obj C, the map F : homC(A,B) → homD(FA, FB) is surjec-tive.

We note that a subcategory C ′ of C is full if and only if the inclusion functor of C ′ on Cis full.

A category C is concrete if and only if there is a faithful functor F : C → Set.

Exercises 2.3

1. Let F : C → D be a full and faithful functor. If A,B ∈ obj C are such that FA ∼= FB, then A ∼= B.

3Natural Transformations


3.1 Definition. Let C and D be categories, and F, F ′ : C → D be two functors. A natural transforma-tion η from F to F ′, denoted η : F→ F ′, is a collection of D-maps ηA : FA→ F ′A, onefor each A ∈ obj C, such that the following diagram commutes for every morphismf : A→ B in the category C

A

f

B

FAηA//

F(f)

F ′A

F′(f)

FBηB// F ′B

(3.1)

If η : F → F ′ is a natural transformation, we call the D-morphism ηA the component of ηcorresponding to A ∈ obj C.

Examples 3.11. Let F : R-mod → R-mod be the functor M 7→ hommod-R

(homR-mod(M,R),R

)considered at

the end of section 2.2. We define a natural transformation η from the identity functor toF. Let M be an object in R-mod, then we define ηM : M → hommod-R(M

∗,R) as m 7→ m,

23

24 Chapter 3. Natural Transformations

where m : M∗ → R is defined as m(φ) = φ(m). We must then check commutativity of

MηM//

f

hommod-R(M∗,R)

hommod-R(f∗,R)

NηN// hommod-R(N

∗,R)

(3.2)

which follows, since for any m ∈M we have that f(m) = m f∗ : N∗ → R.

2. There is a natural isomorphism from the functor π : Top∗ → Grp to the composition

of the functors Top∗ → TopH1−→ Ab → Grp, given by the Hurewicz homomorphism

χ(X,x0) : π(X, x0) → H1(X).

Examples 3.21. Let I be a discrete category, and two functors F, F ′ : I → C given by the collections of

C-objects Fii∈obj I, F ′ii∈obj I respectively. A natural transformation η : F → F ′ is just acollection of C-morphisms η : Fi → F ′ii∈obj I.

2. Let I be the category• //

•

•, and two functors F, F ′ : I → C be represented by the diagrams:

Af//

g

B

C

A ′ f′//

g′

B ′

C ′

(3.3)

respectively. A natural transformation from F to F ′ is a collection of three C-morphismsφ : A→ A ′, ψ : B→ B ′ and ζ : C→ C ′ such that the following diagram commutes:

Af//

g

φ AA

AAAA

A B

ψ

BB

BBBB

BB

C

ζ

AAAA

AAAA

A ′f′//

g′

B ′

C ′

(3.4)

3. Now let I be the category •//

// • , and two functors F, F ′ : I → C be represented by thediagrams:

Af//

g// B A ′

f′//

g′// B ′ (3.5)

3.2. The Functor Category 25

A natural transformation from F to F ′ is then a pair of D-morphismsφ : A→ A ′, ψ : B→ B ′

such that the following squares commute:

Af//

φ

B

ψ

A ′ f′// B ′

Ag//

φ

B

ψ

A ′g′// B ′

(3.6)

Exercises 3.1

1. Let C and D be categories. For each C ∈ obj C, define σC : D → C×D as σC(D) = (C,D), andfor a D-map f : D→ D ′, define σC(f) as (1C, f). Show that σC is a functor. Then, from a C-mapφ : C→ C ′, define a suitable natural transformation σf : σC → σC′ .

3.2 The Functor Category

3.2 Proposition. Let F and G be functors C → D

1. We have that 1F = 1FAA∈obj C is a natural transformation from F to F, called theidentity natural transformation on F.

2. Let η : F1 → F2, χ : F2 → F3 be natural transformations, where F1, F2 and F3 arefunctors C → D. Then we can define a natural transformation χ η by setting(χ η)A = χA ηA. This is called the composition of the natural transformationsη,χ.

Now suppose we have C and D be categories with C small, and let F, F ′ be functorsC → D. We note that a natural transformation η : F→ F ′ is determined by a map

η : obj C → ∪C∈obj C homD(FC, F ′C). (3.7)

Since both obj C and ∪C∈obj C homD(FC, F ′C) are sets, we obtain that the class of all natu-ral transformations from F to F ′ is actually a set.


Then Proposition 3.2 and this remarks allows us to introduce a new category.

3.3 Definition. Let C and D be categories with C small. We define a functor category DC with objectsthe functors from C to D, and the morphisms from a functor F : C → D to F ′ : C → D,the natural transformations from F to F ′. The identity and composition given byProposition 3.2.

3.4 Definition. Remember the category ∆ from Definition 1.10. If C is any category, then the categoryof simplicial objects in C is the category C(∆op).

3.5 Definition. Let C and I be categories with I small. We have a diagonal functor ∆ : C → CI, givenon objects by ∆(C) = FC, where FC : I → C is the constant functor of Example 2.1.3.

The set of natural transformations from one functor to another will then be denotedby homDC(F, F ′). It will be needed in contexts different that those of functor categories.

3.6 Proposition. Let C be a small category,

1. Let S : D → E be a functor. Then there is a functor SC : DC → EC, given on objectsby SC(F) = S F, and if η : F→ F ′ is a morphism in DC, then SC(η) = Sη : S F→S F ′ is given by (Sη)A = S(ηA).

2. Let C ′ be a small category, and T : C ′ → C be a functor. Then there is a functorT∗ : DC → DC′

, given on objects by T∗(F) = F T , and if η : F → F ′ is a morphismin DC, then T∗(η) = ηT is given by (ηT)C = ηTC.

3.7 Proposition. Let C and I be small categories and D be an arbitrary category. Then the categoriesDC×I and (DI)C are isomorphic.

Proof. Step 1. We define a functor Φ : DC×I → (DI)C. For S : C × I → D, let Φ(S) be the

composition C σ−→ (C× I)I SI

−→ DI, where σ is the functor of Exercise 3.2.1.

Step 2. We define a functor Ψ : (DI)C → DC×I. Let T : C → DI. We define Ψ(T) as the

composition C× I(T ,1I)−−−→ DI× I ev−→ D, where ev is the evaluation functor of Exercise 3.2.2.

3.3. Equivalence of Categories 27

Step 3. We check the composition Ψ Φ. For S : C × I → D, we have that (Ψ Φ)(S) =ev (SI σ, 1I). Evaluating at (C, i) ∈ obj C× I, we obtain

[ev (SI σ, 1I)](C, i) = ev((SI σ)(C), i

)(3.8)

= [(SI σ)(C)](i) (3.9)

= (S σC)(i) = S(C, i) (3.10)

and so (Ψ Φ)(S) = S.

Step 4. We check the composition Φ Ψ. For T : C → DI, we have that (Φ Ψ)(T) =(ev (T , 1I)

)I σ. Evaluating at C ∈ obj C, we get

[(ev (T , 1I)

)I σ](C) =(ev (T , 1I)

) σC : I → D (3.11)

and evaluating this at i ∈ obj I, we get

[(ev (T , 1I)

) σC](i) =

(ev (T , 1I)

)(C, i) (3.12)

= ev(T(C), i

)= T(C)(i) (3.13)

hence (Φ Ψ)(T) = T .

Step 5. We leave as an exercise to prove thatΦΨ and ΨΦ are the corresponding identityfunctors.

Exercises 3.2

1. For categories C and D, with D small, we get a functor σ : C → (C×D)D, C 7→ σC, where σC isas in Exercise 3.1.1.

2. For categories C and D, with C small, we have an evaluation functor ev : DC × C → D, definedon objects by ev(F,C) = F(C).

3. Complete the proof of Proposition 3.7.1

4. If D is small, then C(Dop) ∼= (Cop)D ∼= (CD)op.

5. Does the map obj DE → obj(EC)DCgiven by S 7→ SC from Proposition 3.6.1 define a functor?

3.3 Equivalence of Categories


3.8 Definition. Let F, F ′ be functors C → D, and η : F → F ′ a natural transformation. If all compo-nents ηA are isomorphisms, we say that the functors F and F ′ are naturally isomorphic.We denote this as F ∼= F ′.

Natural isomorphism arise frequently in the following way: one observes an isomorphismFA ∼= F ′A which is defined independently of the object A. One then says that the isomor-phism is natural on A.

Note that if C is a small category, then two functors F, F ′ : C → D are naturally iso-morphic if and only if there are isomorphic objects in the functor category DC.

3.9 Definition. Let C and D be categories. We say that C is equivalent to D, if there are functorsF : C → D and F ′ : D → C such that F ′ F ∼= 1C and F F ′ ∼= 1D.

3.4 The Yoneda Lemma

3.10 Definition. Let C be a category. We say that a functor F : C → Set is representable if there isA ∈ obj C such that F ∼= homC(A, −).

3.11 Theorem. (Yoneda Lemma) Let C be any category, F : C → Set a functor, and A ∈ obj C. Thenthere is a bijection FA↔ homSetC

(homC(A, −), F

).

Proof. Given a ∈ A, we define a natural transformation η(a) : homC(A, −) → F. Itscomponent at B ∈ obj C is η(a)B : homC(A,B) → FB sending f 7→ F(f)(a). To see thatη(a) is actually a natural transformation, we check commutativity of the diagram:

B

φ

B ′

homC(A,B)η(a)B

//

homC(A,φ)

FB

F(φ)

homC(A,B ′)η(a)B′

// FB ′

(3.14)

And that follows, since evaluating at f ∈ homC(A,B) it comes down to Fφ(F(f)(a)

)=

F(φ f)(a). Hence we have defined a set map η : FA → homC(A, −). For the inverse, wepropose the map homC(A, −) → FA given by κ : κA(1A). We now show that both mapsare actually inverses to each other.

3.4. The Yoneda Lemma 29

First, given a ∈ FA, we have that η(a)A(1A) = F(1A)(a) = 1FA(a) = a.

Then, given κ ∈ homSetC

(homC(A, −), F

), we want to show that η(κA(1A)) = κ. Let

B ∈ obj C. We use commutativity of

A

f

B

homC(A,A)ηA//

homC(A,f)

FA

F(f)

homC(A,B)ηB// FB

(3.15)

since η(κA(1A)

)B(f) = F(f)

(κA(1A)

)=

(F(f) ηA

)(1A) =

(ηB homC(A, f)

)(1A) =

κB(f). This finishes the proof.

Keeping the hypothesis of the Yoneda Lemma, if A ′ is a C-object, then applying saidLemma to F = homC(A ′, −), we get a bijection between the set homC(A ′,A), and theset of natural transformations from homC(A, −) and homC(A ′, −). We even have thefollowing:

3.12 Corollary. Let A and A ′ be C-objects. Then the functors homC(A, −) and homC(A ′, −) arenaturally isomorphic if and only if A and A ′ are isomorphic.

Proof. We apply the Yoneda Lemma with F = homC(A ′, −).

Let φ ∈ FA = homC(A ′,A) an isomorphism. As in the proof of the Yoneda Lemma,we have a natural transformation η(φ) with component at B ∈ objB being the mapη(φ)B : homC(A,B) → homC(A ′,B) which sends f 7→ f φ. This is a bijection for everyB ∈ obj C (its inverse is η(φ−1)B), so the functors mentioned are naturally isomorphic.

Now, let η : homC(A, −) → homC(A ′, −) be a natural isomorphism. Let φ = ηA(1A),we will show that φ is an isomorphism. Consider the following diagram:

A ′

φ

A

homC(A,A ′)ηA′//

φ∗

homC(A ′,A ′)

φ∗

homC(A,A)ηA// homC(A ′,A)

(3.16)

Since ηA′ is a bijection, there is an f : A→ A ′ such that ηA′(f) = 1A′ . The diagram (3.16)then says that ηA(φ f) = φ. But also ηA(1A) = φ, so that φ f = 1A.


To prove now that f φ = 1A′ , we consider the diagram

A

f

A ′

homC(A,A)ηA//

f∗

homC(A ′,A)

f∗

homC(A,A ′)ηA′// homC(A ′,A ′)

(3.17)

since evaluating at 1A, we get that f φ = ηA′(f) = 1A′ .

Exercises 3.3

1. A contravariant functor F from C to Set is defined to be representable if there is B ∈ obj C suchthat F ∼= homC(−,B). State and prove a Yoneda Lemma for contravariant functors.

2. Let C be any category, and consider the functor homC : Cop × C → Set of Example 2.3. UsingProposition 3.7, to that functor corresponds a functor Cop → SetC. Show that this last functorfull, faithful, and injective in objects. (Note that this, together with Exercise 2.3.1, gives anotherway to prove Corollary 3.12).

Part II

Limits

31

4Limits and Colimits

In this chapter, I will always denote a small category. And, for a functor F : I → C andi ∈ obj I, we will denote the object F(i) by Fi.

4.1 Limits

4.1 Definition. Let F : I → C be a functor and Y an object in C.

1. A source from Y to F is a collection of C-maps δi : Y → Fi, one for each i ∈ obj I.

2. A natural source from Y to F is a source from Y to F such that the following diagramcommutes for all objects i, j in I and maps m : i→ j.

Yδj

;;

;;;;

;δi

FiF(m)

// Fj

(4.1)

Examples 4.11. If A is an initial object in C, then for each C ∈ obj C there is a unique map δC : A → C.

The collection of maps δCC∈obj C is a natural source from A to 1C.

33

34 Chapter 4. Limits and Colimits

2. If the category I is discrete, then any source is a natural source.

3. If I is the category•

• // •

, we saw that a functor F : I → C can be identified with a diagram

in CA

f

Bg// C

(4.2)

A natural source from Y to F may be identified with a pair of morphisms φ : Y → A,ψ : Y → B making commute the following square:

Yφ//

ψ

A

f

Bg// C

(4.3)

4. Let I be the category •//

// • . A natural source to the functor F : I → C

A

f//

g// B . (4.4)

can be identified with a map k : Y → A such that f k = g k.

4.2 Definition. Let F : I → C be a functor. A limiting source of F is a natural source from some C-object Y to F, say δi : Y → Fi, such that for any other natural source δ ′i : Y

′ → Fi

there is a unique C-morphism M : Y ′ → Y such that the diagram

Y ′M

//______

δ′i ;;

;;;;

; Y

δi

Fi

(4.5)

commutes for all i ∈ obj I. In this case, we say that X is a limit of F.

Note that any two limits of F are isomorphic. This limit is sometimes called inverse limitor projective limit.

Examples 4.21. Let p be a prime. The ring of p-adic integers Zp can be defined formally as the set of

sums of the form r0 + r1p + r2p2 + · · · , with 0 ≤ ri < p, and with the natural1 sum and

multiplication.

4.2. Colimits 35

A functor F : Nop → Ab is determined by a collection of abelian groups F(n) = An, one foreach n ∈ N and morphisms Am → An whenever n ≤ m. Let An = Z/pn and Am → Anbe determined by 1 7→ 1. We claim that lim F = Zp. We have maps ηn : Zp → Z/pndefined by r0 + r1p + r2p

2 + · · · 7→ r0 + r1p + r2p2 + · · · + rn−1p

n−1, and they makethe corresponding diagram commute. If we have a collection of maps η′n : X → Z/pnthat make commute diagram 2, then we can define a map X → Zp the following way:By commutativity, given x ∈ X we can find integers ri such that 0 ≤ ri < p and suchthat η′n(x) = r0 + r1p+ r2p2 + · · ·+ rn−1pn−1 for all n ∈ N. Define the map X → Zp asx→ r0 + r1p+ r2p

2 + · · · . This has the commutativity and uniqueness properties requiredfor the limit.

Exercises 4.1

1. Let F : I → C be a functor, δi : Y → Fi a natural source, and φ : Y ′ → Y a C-map. Thenδi φ : Y ′ → Fi is a natural source.

2. Let F : I → C be a functor, δi : Y → Fi a limiting source, and φ,ψ : Y ′ → Y C-maps such thatδi φ = δi ψ for all i ∈ obj I. Then φ = ψ.

3. Let F : I → C be a functor, δi : Y → Fi a limiting source, δ ′i : Y′ → Fi a natural source and

M : Y ′ → Y a C-map such that δi M = δ ′i for all i ∈ obj I. Show that δ ′i : Y′ → Fi is a limiting

source if and only if M is an isomorphism.

4.2 Colimits

4.3 Definition. Let F : I → C be a functor and X an object in C.

1. A sink from F to X is a collection of C-maps εi : Fi → X, one for each i ∈ obj I.

2. A sink from F to X is natural if the following diagram commutes for all objects i, jin I and maps m : i→ j.

FiF(m)

//

εi

;;;;

;;; Fj

εj

X

(4.6)


In other words, a sink from F to X is just a collection of objects in the comma categoryF ↓X, indexed by obj I. If we denote by π the projection functor F ↓X → I, a natural sinkcan be identified with a functor θ : I → F ↓X such that π θ = 1I. Another way that wecan think of a sink from F to X is as a natural transformation from F to ∆(X).

Examples 4.31. If B is a final object in C, then for each C ∈ obj C there is a unique map ηC : C → B, thus

we get a sink from 1C to B. The uniqueness of the maps going to B shows that the sink isnatural.

2. If the category I is discrete, then any sink is a natural sink.

3. Let I be the category category• //

•

•, and the functor F : I → C be represented by the

diagram

Af//

g

B

C

(4.7)

A natural sink from F to Xmay be identified with a pair of morphisms φ : B→ X, ψ : C→ X

making commute the following square:

Af//

g

B

φ

Cψ// X

(4.8)

4. Let I be the category •//

// • , and F be the functor:

A

f//

g// B . (4.9)

Then a natural source from the functor F : I → C to X can be identified with a map l : B→ X

such that l f = l g.

4.4 Definition. Let F : I → C be a functor. A limiting sink for F is a natural sink from F to someC-object X, say εi : Fi → X, such that for any other natural sink ε ′i : Fi → X ′, thereis a unique map M : X→ X ′ such that the diagram

Fiε′i

;;

;;;;

;εi

XM

//______ X ′

(4.10)

4.2. Colimits 37

commutes for all i ∈ obj I. In this case, we say that X is a colimit of F, and we denoteit by colim F.

Note that any two colimits of F are isomorphic.

The colimit is also sometimes called direct limit or inductive limit, in which case it isdenoted as lim

−→F.

A functor F : I → C is called an I-diagram in C.

Examples 4.41. Let F : I → Set be a functor. Let U = (i, x) | i ∈ I, x ∈ F(i) . Then colim F = U/∼, where

∼ is the equivalence relation generated by (i, x) ∼(i ′, F(m)(x)

), where m : i → i ′ is a

morphism of I.

2. We show that any module M is the colimit of its finitely generated submodules. For anyfinitely generated submodule N we have an inclusion map ρN : N → M. They commutewith all inclusionsN1 ≤ N2, so we have a natural sink. Suppose now that we have a moduleX and maps ρ′N : N → X, one for each finitely generated submodule N, forming anothernatural sink. We need to define an f : M→ X. Letm ∈M, and define f(m) = ρ′N(m), whereN is any finitely generated submodule containingm. This is well defined by commutativity,and it is a module homomorphism with the desired properties.

3. Let N be the partially ordered set of positive integers where we set n ≤ m whenever n|m.Let N be the category associated to such poset. We define a functor F : N → Ab by F(n) =

Z/n and if n ≤ m, we set F(n) → F(m) to be determined by 1 → mn

. One has then to provethat F is actually a functor. We claim then colim F = Q/Z. We have maps εn : Z/n→ Q/Zdefined by 1 7→ 1

n, and they form a natural sink from F to Q/Z. Now, if we have another

natural sink from F ε′n : Z/n→ X then we can define a map Q/Z → X by pq7→ ε′q(p), that

makes the corresponding diagram commute and it is unique with respect to such property.

4. We have a functor I ↓ −: I → SCat by sending i to I ↓ i. We calculate colim I ↓ −. Fori ∈ obj I, let εi : I ↓ i → I be the projection functor of Definition 2.2. Then the εi form alimiting cone for I↓−, hence colim I↓− = I. (See [BK72, XI, 2.3])

Exercises 4.2

1. Let C be a small category, and C ∈ obj C. Show that the colimit of the functor homC(C, −): C →Set is the one-point set.


4.3 More Examples

Some particular cases of limits and colimits have special names, we consider them here.

Examples 4.51. If I is a discrete category, then a functor F : I → C can be identified with a collection

Fii∈obj I of objects in C. Then the colimit of F is called the coproduct of the Fi, denoted∐Fi. However, if the set I is finite, say I = 1, . . . ,n we denote

∐Fi as F1 t · · · t Fn.

Dually, the limit of F is called the product of the Fi and is denoted∏Fi. It is denoted as

F1 × · · · × Fn if I is finite.

2. There is only one functor 0 → C, and its colimit is an initial object in C. Dually, its limit isa final object in C.

3. Considering the identity functor 1C : C → C, we have that lim 1C exists if and only if C hasa final object (which is then lim 1C). Similarly, colim 1C exists if and only if C has an initialobject.

4. If I is the category•

• // •

, a limit of a functor F : I → C is called the pullback of the

corresponding diagram. Dually, if I is the category• //

•

•, a limit of a functor F : I → C is

called the pushout of the corresponding diagram.

5. Now let I be the category •//

// • . The limit of the functor F : I → C

A

f//

g// B . (4.11)

can be identified (see Example 4.1) with a certain C-map K → A, and it is called theequalizer of f and g. Dually, the colimit of F can be identified with certain map B→ C andis called the coequalizer of f and g.

6. An important special case of the last construction is the following: Let C be a category witha zero object 0, and f : A→ B a morphism in C. Then we define the cokernel of f, denotedcoker f as the coequalizer of f and 0 : A → B. Also we define the kernel of f, ker f, as theequalizer of f and 0.

Exercises 4.3

1. A sink from F : I → C to X can be identified with a C-morphism∐i∈obj I Fi → X.

2. If B is a final object in a category C with finite products, then C× B ∼= C for all objects C in C.

3. If C is a category with finite products and C,D are objects in C, then C×D ∼= D× C.

4.4. Limit and Colimit as Functors 39

4. If C is a category with finite products and C,D,E are objects in C, then (C×D)×E ∼= C×D×E ∼=C× (D× E).

5. Coequalizers are epics.

6. If C is a category with finite products and C is an object in C, then we get a functor C×−: C →C, sending A 7→ C×A.

4.4 Limit and Colimit as Functors

4.5 Definition. We say that the category C is cocomplete if colim F exists for any functor F : I → C andany I a small category, and that C is complete if lim F exists for any functor F : I → Cand any I a small category.

Remember that a sink from F to X can be identified with a natural transformation from F

to ∆(X). With this viewpoint, we obtain the following result: if colim F exists, then anynatural transformation from F to ∆(X) induces a unique map colim F → X such that thefollowing diagram in CI commutes:

F

::

::::

:

∆(colim F) //___ ∆(X)

(4.12)

Similarly, if lim F exists, then any natural transformation from ∆(Y) to F induces a mapY → lim F such that the following diagram commutes:

∆(Y) //____

::

::::

:∆(lim F)

F

(4.13)

Hence, if C is cocomplete and I is small, we obtain a functor colim : CI → C, definedon the object F to be colim F, and if η : F → F ′ is a natural transformation, we get a mapcolimη : colim F→ colim F ′ induced by the natural transformation F→ F ′ → ∆(colim F ′).The map colimη is also characterized as the only D-map that makes the following squarecommute:

Fiηi

//

εi

F ′i

ε′i

colim Fcolimη

//_____ colim F ′

(4.14)


for all i ∈ obj I.

Similarly if C is complete and I is small, we can define a functor lim : CI → C, sendingF to lim F, and if F→ F ′ is a natural transformation, we get a map lim F→ lim F ′ inducedby the natural transformation ∆(lim F) → F→ F ′.

Another useful remark is that if T : C → D is a functor, and A ∈ obj C, then thecomposition

I∆(A)−−−→ C T−→ D (4.15)

that is T(∆(A)

), is equal to ∆(TA).

Exercises 4.4

1. Let C be a cocomplete category and I be a small category. Form the coproducts∐φ : X→Y F(X),∐

X∈obj C F(X), where φ : X → Y varies over all I-morphisms. Using 4.14, construct maps∐φ F(φ) :

∐φ F(X) →

∐X F(X) and

∐φ 1F(X) :

∐φ F(X) →

∐X F(X). Show that the coequal-

izer of these two maps is colim F.

4.5 Preservation of Limits

Let F : I → C, T : C → D be functors, and Y an object in C. If δi : Y → Fi is a sourcefrom Y to F, then T δi : T(Y) → T(Fi) is a source from T(Y) to T F. Furthermore, if theoriginal source is natural, then the resulting source is natural as well.

Suppose then that δi : Y → Fi is a limiting source, that is, Y = lim F. Since T δiis a natural source from T(Y) to T F, by the definition of limit there is a unique mapM : T(lim F) → lim(T F) making the following triangle commute for all i ∈ obj I.

T(lim F) M//___

Tδi

::::

:::

lim(T F)

T F

(4.16)

The following definition considers the case in which the map M is an isomorphism.

4.5. Preservation of Limits 41

4.6 Definition. We say that the functor T : C → D preserves limits, if for any functor F : I → C andlimiting source δi : Y → Fi from Y to F, then T δi : T(Y) → T(Fi) is a limitingsource from T(Y) to T F.

Theorem 5.6 will give a condition on F that ensures preservation of limits.


5Universals and Adjoints

5.1 Universals

5.1 Definition. Let C and D be categories, F : C → D be a functor, and B ∈ obj D. Then a universalfrom B to F is a pair (U,u) where U ∈ obj C and u : B→ FU is a D-map, such that ifh : B→ FU ′ is any D-map with U ′ ∈ obj C, then there is a unique C-map m : U→ U ′

such that the following diagram commutes:

FU

F(m)

U

m

B

u 77oooooo

h ''OOOOOO

FU ′ U ′

(5.1)

That is, any map of the form h : B → FU ′ can be factored through u. Also, it intuitivelymeans that in order to go out of U to U ′ in C it is enough to go from B to FU ′ in D. Andit also means that (U,u) is an initial object in the comma category B↓F.

Examples 5.11. Consider the forgetful functor F : Ab → Set and X ∈ obj Set. Then a universal from X to F

is the pair (ZX,u), where ZX is the free abelian group with base X and u is the inclusionu : X→ F(ZX). Clearly, if A is an abelian group, and if

h : X→ FA (5.2)

43

44 Chapter 5. Universals and Adjoints

is a map of sets, then there is a unique map of abelian groups

m : ZX→ A (5.3)

that makes the following diagram commute:

F(ZX)

F(m)

ZX

m

X

u 77nnnnnn

h ((QQQQQQQ

FA A

(5.4)

That is, in order to define a homomorphism from ZX to the abelian group A, it is enoughto define a set map from X to the set FA, the underlying set of the group A.

2. Let I be an small category, C an arbitrary category, and ∆ : C → CI the diagonal functor. IfF ∈ obj CI, a CI-map u : F → ∆(U), with U ∈ obj C is the same as a natural sink from F toU. We have that (U,u) is a universal from F to ∆ precisely when the sink is limiting. Thediagram then looks like:

∆(U)

∆(m)

U

m

F

u 77oooooo

h''OOOOOO

∆(U ′) U ′

(5.5)

With the setup of Definition 5.1, if (U,u) and (U ′,u ′) are universals from B to F, thenthere is a unique C-isomorphism m : U→ U ′ such that u ′ = F(h) u.

5.2 Definition. Let C and D be categories, F : C → D be a functor, and A ∈ obj D. Then a universalfrom F to A is a pair (V, v) where V ∈ obj C and v : FV → A is a D-map, such that ifk : FV ′ → A is any D-map with V ′ ∈ obj C, then there is a unique C-map m : V ′ → V

such that the following diagram commutes:

FV ′k

''OOOOOO

F(m)

V ′

m

A

FVv

77ooooooV

(5.6)

5.2. Adjoint Functors 45

5.2 Adjoint Functors

5.3 Definition. Let C and D be categories, and T : C → D, S : D → C be functors. We say that S isleft adjoint to T if for all A ∈ obj D, B ∈ obj C we have a bijective map ηA,B fromhomC(SA,B) to homD(A, TB) which is natural in A and B. We denote this as S a T ,and the map η is called the adjugant of the adjunction S a T .

In other words, this means that for any A ∈ obj D, the functors homC(SA, −) andhomD(A, T−) are naturally isomorphic, and that for any B ∈ obj C, the contravariantfunctors homC(S−,B) and homD(−, TB) are naturally isomorphic, that is, the followingdiagram commutes for all A ∈ obj D and all maps f : B→ B ′ in C

B

f

B ′

homC(SA,B)ηA,B

//

f∗

homD(A, TB)

(Tf)∗

homC(SA,B ′)ηA,B′

// hom(A, TB ′)

(5.7)

and the following diagram commutes for all B ∈ obj C and all maps g : A→ A ′ in D

A

g

A ′

homC(SA,B)ηA,B

// hom(A, TB)

hom(SA ′,B)ηA′ ,B

//

(Sg)∗

OO

hom(A ′, TB)

g∗

OO

(5.8)

Example 5.2Let C be a cocomplete category. Then for any small category I we have that colim : CI → C isleft adjoint to the diagonal functor ∆ : C → CI. And if C is a complete category, we have thatlim : CI → C is right adjoint to the diagonal.

Now, as in the situation of Definition 5.3, suppose that S and T are functors such thatS a T . For each object A of D, let εA : A → TSA be εA = ηA,SA(1SA). It can be proventhat the collection of maps ε = εA gives a natural transformation 1 → TS. Similarly,for each B ∈ obj C, let χB = η−1

TB,B(1TB) : STB → B, then χ = χA can be shown to be anatural transformation ST → 1.

5.4 Definition. If S and T are functors such that S a T , the natural transformation ε described in theprevious paragraph is called the unit of the adjunction S a T . The natural transfor-mation χ is called the counit.


5.5 Proposition. If S and T are functors such that S a T with unit ε : 1 → TS and counit χ : ST → 1,then the compositions:

SSε−→ STS

χS−→ S, (5.9)

TεT−→ TST

Tχ−−→ T (5.10)

are equal to the respective identities. Conversely, let S : D → C and T : C → D befunctors such that there are natural transformations ε : 1 → TS, χ : ST → 1 such thatthe compositions 5.9 are equal to the identity. Then S a T , and ε is the unit of theadjunction and χ the counit. The adjugant ηA,B can be obtained as follows: ηA,B(h)is the composition

BεB−→ TSB

Th−−→ TA (5.11)

5.6 Theorem. Let C and D be categories, and T : C → D, S : D → C be functors such that S a T .Then

1. S preserves all colimits.

2. T preserves all limits.

Proof. We show that T preserves all limits. Let F : I → C be a functor with limiting sourceδi : X→ Fi, we need to show that Tδi : TX→ TFi is a limiting source. Let ρi : Z→ TFi

be a natural source, we want to show there is a unique D-morphism M : Z → TX thatmakes the following diagram commute

ZM

//______

ρi

;;;;

;;; TX

Tδi

TFi

(5.12)

Consider the adjunction map ηZ,Fi: homC(SZ, Fi) → homD(Z, TFi). We get maps η−1

Z,Fi(ρi) : SZ→

Fi, we want to show they form a natural source. Letm : i→ j be an I-morphism, we wantto prove that the diagram

SZη−1

Z,Fi(ρj)

;;

;;;;

;η−1

Z,Fi(ρi)

FiF(m)

// Fj

(5.13)

5.2. Adjoint Functors 47

is commutative. But this follows from the commutativity of the diagram

i

m

j

Fi

F(m)

Fj

homC(SZ, Fi)ηZ,Fi

//

F(m)∗

homD(Z, TFi)

TF(m)∗

homC(SZ, Fj)ηZ,Fj

// homD(Z, TFj)

(5.14)

since ρj = TF(m)ρi. We obtain then that there is a unique mapM ′ : SZ→ X making thediagram

SZM′

//______

η−1Z,Fi

(ρi)

;;;;

;;; X

δi

Fi

(5.15)

commute for all i ∈ obj I. From the adjunction map ηZ,X : homC(SZ,X) → homD(Z, TX),let M = ηZ,X(M ′), we want now to prove that the diagram (5.12) commutes for eachi ∈ obj I. This follows from commutativity of the diagram

X

δi

Fi

homC(SZ,X)ηZ,X

//

(δi)∗

homD(Z, TX)

(Tδi)∗

homC(SZ, Fi)ηZ,Fi

// homD(Z, TFi)

(5.16)

and evaluating at M ′ in the upper left corner.


6More on Limits

6.1 Limits in a Functor Category

6.1 Theorem. Let I, C be small categories, D be a cocomplete category, and T : I → DC. Then thefunctor T has a colimit, which can be calculated point-wise.

Proof. Step 1. We prove that for each C ∈ obj C, there is a functor I → D defined inobjects by i 7→ T(i)(C), which we will denote as T(−)(C): Let m : i → j be a map in I,we need to define a D-map T(m)(C) : T(i)(C) → T(j)(C). Then T(m) : T(i) → T(j) is anatural transformation of functors C → D, we define T(m)(C) as the C-component of thenatural transformation T(m). Using the definition of natural transformation, it is clearthat it preserves compositions and so we have defined a functor T(−)(C).

Step 2. We show there is a functor C → D defined in objects as C → coliml T(l)(C),which we will denote as coliml T(l)(−): Let f : C → C ′ be a C-morphism, we want todefine a natural transformation T(−)(f) : T(−)(C) → T(−)(C ′) : I → D. The i-componentof that is precisely T(i)(f). This is indeed a natural transformation, since given an I-mapm : i → j, we get the following commutative diagram, using that T(m) : T(i) → T(j) is anatural transformation:

C

f

C ′

T(i)(C)T(m)(C)

//

T(i)(f)

T(j)(C)

T(j)(f)

T(i)(C ′)T(m)(C′)

// T(j)(C ′)

(6.1)

49

50 Chapter 6. More on Limits

which we can interpret as:

i

m

j

T(i)(C)T(i)(f)

//

T(m)(C)

T(i)(C ′)

T(m)(C′)

T(j)(C ′)T(j)(f)

// T(j)(C ′)

(6.2)

Hence the natural transformation T(−)(f) : T(−)(C) → T(−)(C ′) induces a D-morphismcoliml T(l)(C) → coliml T(l)(C ′), which we will denote as coliml T(l)(f).

For the record, we note that this morphism has the property of being the unique D-mapthat makes the following diagram commute for all i ∈ obj I.

T(i)(C)T(i)(f)

//

εCi

T(i)(C ′)

εC′i

coliml T(l)(C)coliml T(l)(f)

// coliml T(l)(C ′)

(6.3)

Step 3. We construct a sink εi from T : I → DC to the DC-object coliml T(l)(−): Wedefine a collection of DC-morphisms T(i) → coliml T(l)(−), one for each i ∈ obj I. ForC ∈ obj C, we set the C-component of εi as the map εCi of diagram (6.3). This is a naturaltransformation, since the diagram:

C

f

C ′

T(i)(C)εC

i//

T(i)(f)

coliml T(l)(C)

coliml T(l)(f)

T(i)(C ′)εC′

i// coliml T(l)(C ′)

(6.4)

is diagram (6.3). So εi is a sink.

Step 4. We show that εi is a natural sink: Let m : i → j be an I-morphism, we need toshow commutativity of

T(i)T(m)

//

εi

9999

999

T(j)

εj

coliml T(l)(−)

(6.5)

which follows from commutativity of

T(i)(C)T(m)(C)

//

εCi

9999

999

T(j)(C)

εCj

coliml T(l)(C)

(6.6)

6.1. Limits in a Functor Category 51

for each C ∈ obj C, since this is precisely the condition that defines the maps εCi .

Step 5. We show that the maps εi : T(i) → coliml T(l)(−) form a limiting sink, that is, thefunctor coliml T(l)(−) : C → D is the colimit of T : I → DC. Let ε ′i : T(i) → Z be a naturalsink. Evaluating at C ∈ obj C, we get a natural sink (ε ′i)

C : T(i)(C) → Z(C), and so weobtain a map MC making the following diagram commute:

T(i)(C)

(ε′i)C

99

9999

9εC

i

coliml T(l)(C)MC//__ Z(C)

(6.7)

We want to show that theMC are the components of a natural transformation coliml T(l)(−) →Z, that is, that the following diagram is commutative for all C-maps f : C→ C ′.

C

f

C ′

coliml T(l)(C)MC

//

coliml T(l)(f)

Z(C)

Z(f)

coliml T(l)(C ′)MC′

// Z(C ′)

(6.8)

Since ε ′i : T(i) → Z is a natural transformation, we have commutativity of

C

f

C ′

T(i)(C)(ε′i)C

//

T(i)(f)

Z(C)

Z(f)

T(i)(C ′)(ε′i)C′

// Z(C ′)

(6.9)

which combined with diagrams (6.3) and (6.7) results in the diagram of Figure 6.1.

From there, we get that both Z(f) MC and MC′ coliml T(l)(f), when in the place ofthe dotted arrow, make the following diagram commute for all C ∈ obj C:

T(i)(C)

Z(f)(ε′i)C

99

9999

9εC

i

coliml T(l)(C) //__ Z(C ′)

(6.11)

By uniqueness, given that the εCi form a limiting sink, we have that Z(f) MC = MC′ coliml T(l)(f), and hence we obtain the result.

6.2 Corollary. Suppose we have a functor S : C × I → D. We can interpret S as a functor withparameter i.


coliml T(l)(C)

MCxxppppppppppp

coliml T(l)(f)

T(i)(C)

εCi

44hhhhhhhhhhhhhhhhhhh

(ε′i)C

//

T(i)(f)

Z(C)

Z(f)

T(i)(C ′)

εC′i **VVVVVVVVVVVVVVVVVVV

(ε′i)C′

// Z(C ′)

coliml T(l)(C ′)

MC′ffNNNNNNNNNNN

(6.10)

Figure 6.1: Diagram

6.3 Corollary. If D is a complete category, C is a small category and F1, . . . , Fn are functors C → D,then we have a product functor,

∏ni=1 Fj : C → D, defined as (

∏Fj)(A) =

∏Fj(A),

for A ∈ obj C.

6.2 Ends

6.4 Definition. Let C and D be categories, and S, T be functors Cop × C → D. A dinatural transfor-mation α : S → T is a collection of D-maps, αA : S(A,A) → T(A,A), indexed by theobjects of C, such that the following diagram commutes for every C-map f : A→ B.

S(A,A)αA// T(A,A)

T(1A,f)

%%KKKKKKKKK

S(B,A)

S(f,1A)99sssssssss

S(1B,f)%%K

KKKKKKKKT(A,B)

S(B,B)αB// T(B,B)

T(f,1B)

99sssssssss

(6.12)

Example 6.1

6.2. Ends 53

If η : S → T is a natural transformation, then η(C,C) is a dinatural transformation. Considerthe following diagram:

S(B,B)η(B,B)

// T(B,B)

T(f,1B)

S(B,A)η(B,A)

//

S(f,f)

S(f,1A)

S(1B,f)ddJJJJJJJJJ

T(B,A)

T(1B ,f)99ttttttttt

T(f,f)

S(A,B)η(A,B)

// T(A,B)

S(A,A)

S(1A,f)

::ttttttttt η(A,A)// T(A,A)

T(1A,f)eeJJJJJJJJJ

(6.13)

Then we have:

T(1A, f) η(A,A) S(f, 1A) = η(A,B) S(1A, f) S(f, 1A) (6.14)

= η(A,B) S(f, f) (6.15)

= T(f, f) η(B,A) (6.16)

= T(f, 1B) T(1B, f) η(B,A) (6.17)

= T(f, 1B) η(B,B) S(1B, f) (6.18)

6.5 Definition. A dinatural sink α from a functor S : Cop × C → D to X ∈ obj D is a dinaturaltransformation from the functor S to the constant functor Cop×C → D with value X.In detail, it is a collection of D-maps αC : S(C,C) → X, indexed by the objects of C,such that for every f : A→ B the following diagram

S(B,A)S(1B,f)

//

S(f,1A)

S(B,B)

αB

S(A,A)αA

// X

(6.19)

commutes.

Example 6.2Let X ∈ obj Set fixed. For each A ∈ obj Set, we have a set map eA : homSet(A,X) × A → X,given by evaluation, that is, eA(f,a) = f(a). The maps eA form a dinatural sink from thefunctor homSet(−,X) × (−) : Setop × Set → Set to the set X, since for every set map φ : A → B


the following square commutes:

homSet(B,X)×A1×φ//

φ∗×1A

homSet(B,X)× B

αB

homSet(A,X)×AαA

// X

(6.20)

6.6 Definition. A dinatural source β from Y ∈ obj D to the functor S : Cop×C → D it is a collection ofD-maps βC : Y → S(C,C)C∈obj C such that for every f : A→ B the following diagramcommutes

YβB

//

βA

S(B,B)

S(f,1B)

S(A,A)S(1A,f)

// S(A,B)

(6.21)

6.7 Definition. An end of a functor S : Cop × C → D is a dinatural source from an object Y to S, suchthat for every dinatural source β ′ from S to some object Y ′ there is a unique D-mapY ′ → Y such that the following diagram commutes.

Y ′M

//______

β′C

::::

::: Y

βC

S(C,C)

(6.22)

for every C ∈ obj C. We denote this as

Y =

∫C

S(C,C) (6.23)

and we also say that the dinatural source β is ending.

Example 6.3Let C be a small category, and F, F ′ : C → D be two functors. We can then consider the func-tor Cop × C → Set given by (A,B) 7→ homD(FA, F ′B). We claim that

∫A

homD(FA, F ′A) =

homDC(F, F ′), the set of natural transformations from F to F ′. Let βC : homDC(F, F ′) →homD(FC, F ′C) be given by βC(η) = ηC. Then β is dinatural, since the condition of

6.2. Ends 55

square (6.21) says in this case

homDC(F, F ′)βB//

βA

homD(FB, F ′B)

(Ff)∗

homD(FA, F ′A)(F′f)∗

// homD(FA, F ′B)

(6.24)

which holds, since for η ∈ homDC(F, F ′), we have that ηB F(f) = F ′(f) ηA. If β ′C : Z →homD(FC, F ′C) is another dinatural source with Z a set, then for each z ∈ Z one gets a D-map β ′C(z) : FC → F ′C, and the collection β ′C(z)C∈obj C is a natural transformation F → F ′.Hence we have a map M : Z → homDC(F, F ′) given by z 7→ β ′C(z)C∈obj C, which satisfies thecommutativity condition 6.22.

6.8 Theorem. Let γ : S→ S ′ be a natural transformation between functors Cop ×C → D with ends.Then there is a unique D-map

∫C γC,C :

∫C S→

∫C S

′ such that the following diagramcommutes for every C ∈ obj C.

∫C S(C,C)

∫C γ//___

βC

∫C S

′(C,C)

β′C

S(C,C)γC,C

// S ′(C,C)

(6.25)

Proof. The collection of compositions ∫C S

βC−−→ S(C,C)γC,C−−−→ S ′(C,C)C∈obj C, is a di-

natural source, because of the diagram:

∫S

βB//

βA

S(B,B)γB,B

//

S(f,1B)

S ′(B,B)

S′(f,1B)

S(A,A)

γA,A

S(1A,f)// S(A,B)

γA,B

((PPPPPPPPPPPP

S ′(A,A)S(1A,f)

// S ′(A,B)

(6.26)

where the two trapezoids commute because of the naturality of γ. By definition of∫S ′,

there is a unique map M :∫S →

∫S ′ making the following triangle commute for all


C ∈ obj C, ∫S

M//_____

γβC

9999

999

∫S ′

β′C

S ′(C,C)

(6.27)

then let M =∫g.

Exercises 6.1

1. Let F : C → D be a functor, and S : Cop × C → D be the functor that is the compositionof projection onto the second factor from Cop × C, with F (that is, S(A,B) = F(B)). Then∫CS(C,C) = lim F.

2. Let S : Cop×C → D be a functor, βC : Y → S(C,C)C∈obj C be a dinatural source, and φ : Y ′ → Y

be a C-map. Then βC φ : Y ′ → S(C,C)C∈obj C is a dinatural source.

3. Let S : Cop ×C → D be a functor, βC : Y → S(C,C)C∈obj C be an ending source, and φ,ψ : Y ′ →Y C-maps such that βC φ = βC ψ for all C ∈ obj C. Then φ = ψ.

6.3 Ends in a Functor Category

6.9 Theorem. Let C, E be small categories, D complete and T : Cop × C → DE be a functor. Then Thas an end, which can be calculated pointwise.

Proof. Step 1. We show that for each E ∈ obj E, there is a functor Cop × C → D, definedon objects as (A,B) 7→ T(A,B)(E). Let (φ,ψ) : (A,B) → (A ′,B ′) be a map in Cop × C.Then T(φ,ψ) : T(A,B) → T(A ′,B ′) is a natural transformation, so we define T(φ,ψ)(E)as the E-component of it.

Step 2. We show that there is a functor E → D, defined on objects as E 7→∫C T(C,C)(E),

which we will denote as∫C T(C,C)(E): First, for an E-morphism f : E → E ′, we want a

natural transformation T(−, −)(E) → T(−, −)(E ′), that is, for each (A,B) ∈ obj Cop × C,

6.3. Ends in a Functor Category 57

we need a map T(A,B)(E) → T(A,B)(E ′). Let it be T(A,B)(f). We need to show it isnatural, that is

(A,B)

(φ,ψ)

(A ′,B ′)

T(A,B)(E)T(A,B)(f)

//

T(φ,ψ)(E′)

T(A,B)(E ′)

T(φ,ψ)(E)

T(A ′,B ′)(E)T(A′,B′)(f)

// T(A ′,B ′)(E ′)

(6.28)

but this follows from the diagram

E

f

E ′

T(A,B)(E)T(φ,ψ)(E)

//

T(A,B)(f)

T(A ′,B ′)(E)

T(A′,B′)(f)

T(A,B)(E ′)T(φ,ψ)(E′)

// T(A ′,B ′)(E ′)

(6.29)

We note that, by Theorem 6.8, the natural transformation T(−, −)(f) induces a D-map

∫C T(C,C)(f) :

∫C T(C,C)(E) →

∫C T(C,C)(E ′), which has the property of being the

unique map that makes the following diagram commute for all A ∈ obj C.

∫C T(C,C)(E)

∫C T(C,C)(f)

//

βEA

∫C T(C,C)(E ′)

βE′A

T(A,A)(E)T(A,A)(f)

// T(A,A)(E ′)

(6.30)

Step 3. We construct a dinatural source from the DE-object∫C T(C,C)(−) to T . We

need to define a collection of DE-morphisms βA :∫C T(C,C)(−) → T(A,A), one for each

A ∈ obj C. We set βEA as in diagram (6.30). That βEAA∈obj C is a natural transformationfollows from that diagram. We now show that it is dinatural, that is, that for every C-mapf : A→ B the following square commutes:∫

C T(C,C)(−)βB//

βA

T(B,B)

T(f,1B)

T(A,A)T(1A,f)

// T(A,B)

(6.31)

Evaluating at E, we get

∫C T(C,C)(E)

βEB

//

βEA

T(B,B)(E)

T(f,1B)(E)

T(A,A)(E)T(1A,f)(E)

// T(A,B)(E)

(6.32)


But this is precisely the condition that defines∫C T(C,C)(E), and so the diagram (6.31)

commutes.

Step 4. This dinatural source is an ending source. Let β ′A : Z → T(A,A) be a dinaturalsource, with Z ∈ obj DE. Evaluating at E, we obtain a map ME that makes the followingdiagram commute:

Z(E)ME//__

(β′A)E

99

9999

9

∫C T(C,C)(E)

βEA

T(A,A)(E)

(6.33)

We want to show that the collection of maps MEE∈obj E is a natural transformationZ→

∫C T(C,C)(−), that is,

E

f

E ′

Z(E)ME

//

Z(f)

∫C T(C,C)(E)

∫C T(C,C)(f)

Z(E ′)ME′

//∫C T(C,C)(E ′)

(6.34)

Since β ′A : Z → T(A,A) is a natural transformation, we have commutativity of the smallrectangle in the following diagram:∫

C T(C,C)(E)

βEA

**VVVVVVVVVVVVVVVVVV

∫C T(C,C)(f)

Z(E)

ME

ffMMMMMMMMMMM

(β′A)E

//

Z(f)

T(A,A)(E)

T(A,A)(f)

Z(E ′)

ME′

xxqqqqqqqqqq

(β′A)E′

// T(A,A)(E ′)

∫C T(C,C)(E ′)

βE′A

44hhhhhhhhhhhhhhhhhh

(6.35)

The triangles are instances of (6.33), and the biggest trapezoid is (6.30). Commutativity ofthe smallest trapezoid is not immediate, but it follows from the fact that both ME′ Z(f)and

∫C T(C,C)(f) ME complete the following commutative triangle, when in place of

the dotted arrow:Z(E) //__

T(A,A)(f)(β′A)E

99

9999

9

∫C T(C,C)(E ′)

βE′A

T(A,A)(E ′)

(6.36)

6.4. Iterated Ends 59

6.4 Iterated Ends

6.10 Lemma. Consider the following diagram:

• 5 //

8

•

6

•9

??

1 //

4

•10

??

2

• 7 // •

• 3 //

12

??

•11

??

(6.37)

Suppose that all faces of the cube are commutative squares, except maybe the frontand back. Then

1. if the front face commutes, then 7 8 9 = 6 5 9,

2. if the back face commutes, then 11 2 1 = 11 3 4.

Proof. Exercise.

6.11 Lemma. Let C, E be small categories, D an arbitrary category, and T : Cop × C× Eop × E → Dbe a functor. Let κC,E : Y → T(C,C,E,E) that is a dinatural source on C for E fixed,and a dinatural source on E for C fixed. Then κ is also a dinatural source when T isconsidered as a functor (C× E)op × (C× E) → D.

Proof. Let f : A → B, h : E → F be morphisms in C, E respectively. We must showcommutativity of the square:

YκB,F

//

κA,E

T(B,B, F, F)

T(f,1B,h,1F)

T(A,A,E,E)T(1A,f,1E,h)

// T(A,B,E, F)

(6.38)

Since κC,E is dinatural in C, when the second variable is fixed we have commutativity ofthe square

YκB,F

//

κA,F

T(B,B, F, F)

T(f,1B,1F,1F)

T(A,A, F, F)T(1A,f,1F,1F)

// T(A,B, F, F)

(6.39)


and since κC,E is dinatural in E, when the first variable is fixed we have commutativity ofthe square

YκA,F

//

κA,E

T(A,A, F, F)

T(1A,1A,h,1F)

T(A,A,E,E)T(1A,1A,1E,h)

// T(A,A,E, F)

(6.40)

We have then

T(1A, f, 1E,h) κA,E = T(1A, f, 1E, 1F) T(1A, 1A, 1E,h) κA,E (6.41)

= T(1A, f, 1E, 1F) T(1A, 1A,h, 1F) κA,F (6.42)

= T(1A, 1B,h, 1F) T(1A, f, 1F, 1F) κA,F (6.43)

= T(1A, 1B,h, 1F) T(f, 1B, 1F, 1F) κB,F (6.44)

= T(f, 1B,h, 1F) κB,F (6.45)

6.12 Theorem. Let C, E be small categories, D complete and T : Cop ×C× Eop × E → D be a functor.Then there is an isomorphism∫

(C,E)

T(C,C,E,E) →∫C

(∫E

T(C,C,E,E))

(6.46)

where in the first integral we have interpreted T as a functor (C×E)op× (C×E) → D

Proof. For each (A,B) ∈ obj Cop × C, we have the ending source (in E)

βA,B,E :

∫E

T(A,B,E,E) → T(A,B,E,E), (6.47)

and we also have the ending source in C:

ρC :

∫C

(∫E

T(C,C,E,E))→

∫E

T(C,C,E,E). (6.48)

Hence we get a collection of maps κC,E, indexed by the objects of C× E, given by thecompositions:

κC,E :

∫C

(∫E

T(C,C,E,E)) ρC−−→

∫E

T(C,C,E,E)βC,C,E−−−−→ T(C,C,E,E), (6.49)

which is a dinatural source in E by Exercise 6.1.2,

6.4. Iterated Ends 61

Let f : A→ B be a C-map, and consider the square

∫C

(∫E T(C,C,E,E)

) ρA//

ρB

∫E T(A,A,E,E)

∫E T(1A,f,E,E)

∫E T(B,B,E,E)

∫E T(f,1B,E,E)

//∫E T(A,B,E,E)

(6.50)

which commutes, since ρC is a dinatural source, and the square

∫C

(∫E T(C,C,E,E)

) κA,E//

κB,E

T(A,A,E,E)

T(1A,f,1E,1E)

T(B,B,E,E)T(f,1B,1E,1E)

// T(A,B,E,E)

(6.51)

which we want to prove commutative. Consider the cubical diagram, that has as frontface the square (6.50) and as back face the square (6.51).

∫C

(∫E T(C,C,E,E)

)ρA //

ρB

∫E T(A,A,E,E)

∫E T(1A,f,E,E)

∫E T(B,B,E,E)

∫E T(f,1B,E,E) //

∫E T(A,B,E,E)

∫C

(∫E T(C,C,E,E)

)κA,E //

κB,E

T(A,A,E,E)

T(1A,f,1E,1E)

T(B,B,E,E) T(f,1B,1E,1E) // T(A,B,E,E)

1∫C

∫E T

??βA,A,E

AA

βB,B,E

??βA,B,E

AA

(6.52)

In this cube, the bottom and right faces commute by Theorem 6.8, and the top and leftfaces do by definition of κ (6.49). By Lemma 6.10, the back face commutes, and so κC,E

is a dinatural source in C. By Lemma 6.11, κ is a dinatural source in both variables.We show now that κ is a limiting source. Let αC,E : Z → T(C,C,E,E) be a dinaturalsource (in both variables). Since α is a dinatural source for C fixed, there is a unique mapµC : Z→

∫E T(C,C,E,E) that makes the following diagram commute

ZµC//___

αC,E

9999

999

∫E T(C,C,E,E)

βC,C,E

T(C,C,E,E)

. (6.53)


For f : A→ B, consider now the cubical diagram

Z µA //

µB

∫E T(A,A,E,E)

∫E T(1A,f,1E,1E)

∫E T(B,B,E,E)

∫E T(f,1B,1E,1E) //

∫E T(A,B,E,E)

Z αA,E //

αB,E

T(A,A,E,E)

T(1A,f,1E,1E)

T(B,B,E,E) T(f,1B,1E,1E) // T(A,B,E,E)

1Z

??βA,A,E

CC

βB,B,E

??βA,B,E

CC

(6.54)

Bottom and right faces are the same of the cube (6.52), so they commute, and top andleft faces commute since they are instances of (6.53). Since the back face commutes bydinaturality of α, by Lemma 6.10 we have that

βA,B,E ∫E

T(1A, f, 1E, 1E) µA = βA,B,E ∫E

T(f, 1B, 1E, 1E) µB (6.55)

for all f : A→ B and E ∈ obj E. Since βA,B,EE∈obj E is an ending source, then∫E

T(1A, f, 1E, 1E) µA =

∫E

T(f, 1B, 1E, 1E) µB (6.56)

by Exercise 6.1.3. Hence µC is a dinatural source, and so there is a unique map M : Z→∫C

(∫E T(C,C,E,E)

)that makes the following diagram commute

Z

M

αC,E

,,YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

µC))SSSSSSSSSSSSSSSSS

∫E T(C,C,E,E) βC,C,E // T(C,C,E,E)

∫C

(∫E T(C,C,E,E)

)ρC

55llllllllllllllκC,E

22eeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

(6.57)

which finishes the proof that κC,E is and ending source.

Exercises 6.2

6.5. Coends 63

1. Prove Lemma 6.10.

2. Considering again the cube (6.37) from Lemma 6.10, if all faces commute except the bottom,then 7 12 4 = 11 3 4.

6.5 Coends

6.13 Definition. A dinatural source from Y ∈ obj∂ to a functor S : Cop × C → D is a dinatural trans-formation from the constant functor Cop ×C → D with value Y to S. In other words,it is a collection of maps


Part III

Extras

65

7Abelian Categories

7.1 Definition

7.1 Definition. The category C is an Ab-category if every set homC(A,B) has a structure of abeliangroup in such a way that composition is distributive over the additive structure. Thatis, for any maps f : A→ B, h1,h2 : B→ C and k : C→ D, we have that k(h1+h2)f =k h1 f+ k h2 f.A functor between Ab-categories F : C → D is called additive if for each pair of objectsA,B ∈ obj C we have that F : homC(A,B) → homD(FA, FB) is a group homomor-phism.

Examples 7.11. The category Ab is an Ab-category, where we define, for f,hhomAb(A,B) the sum f+ h as

the map x 7→ (f+ h)(x) = f(x) + h(x). Similarly, R-mod is an Ab-category any ring R.

2. The definition of f+ h as x 7→ (f+ h)(x) = f(x) + h(x) does not work in the category Rng,since f+ h does not preserve in general the product in rings.

3. Ab-categories with only one object, can be identified with rings. If R is a ring, denote by Rthe associated Ab-category with one object ∗. Then an additive functor F : R → Ab can beidentified with an structure of left R-module on the abelian group F(∗) = M, since F mapsr ∈ R to the abelian group map r : M→M, m 7→ rm.

7.2 Definition. An additive category is an Ab-category with a zero object, and a product for any pairof objects.

67

68 Chapter 7. Abelian Categories

In an additive category, the zero morphism in hom(A,B) is precisely the zero element.

7.3 Definition. An abelian category C is an additive category such that

1. Every morphism in C has a kernel and a cokernel.

2. Every monic in C is the kernel of its cokernel.

3. Every epic in C is the cokernel of its kernel.

1 2

In an abelian category, monics and epics are usually called respectively monomor-phisms and epimorphisms.

3 4

7.2 Chain Complexes

7.4 Definition. Let I be a category with obj I = Z and no morphisms but the identities. Let C be anabelian category. Then the category CI is called the category of graded objects of C.It can be identified with the category such that the objects are collections Cii∈Z ofobjects in C and a morphism Ci → Di is a collection of maps φi : Ci → Di. Wedenote this category as Gr(C).

7.5 Definition. If C is an abelian category, a chain complex C in C is a collection Cii∈Z of objectsin C and maps (called differentials) di : Ci → Ci−1i∈Z such that dn+1dn = 0 for alln ∈ Z. A chain map between the chain complexes C and C ′ is a collection of mapsφi : Ci → C ′i such that the following diagram commutes for all n ∈ Z

Cn+1dn+1

//

φn+1

Cn

φn

C ′n+1

d′n+1// C ′n

1FiXme: EXAMPLE: RING IS NOT ABELIAN2FiXme: If C is abelian, then CI is abelian3FiXme: EXAMPLES? MAYBE MACKEY FUNCTORS. IE THE CATEGORY OF MACKEY FUNCTORS

OVER A GROUP G IS ABELIAN4FiXme: Define the zero functor

7.3. Exact Sequences 69

7.6 Proposition. Chain complexes and chain maps in C form an abelian category which we denoteCh(C).

Note that if F : C → D is a functor, we can immediately define a functor Gr(C) → Gr(D),and if F is also additive, we can even define a functor Ch(C) → Ch(D), which we alsodenote by F. Also note that we have a forgetful functor Ch(C) → Gr(C) and a functorGr(C) → Ch(C) that sends a graded object to a complex with all differentials to be zero.

We usually have chain complexes in which Ci = 0 for i < 0, they are called positivechain complexes. We could also have a complex for which Ci = 0 if i > 0, but if thisis the case, we call Ci = C−i and d−i : C−i → C−i−1 becomes di : Ci → Ci+1. We saythen that Ci is a cochain complex. The category of positive chain complexes is denotedCh≥0(C) and the category of cochain complexes is denoted Ch≥0(C).

5

7.3 Exact Sequences

7.7 Definition. Let f : A → B a morphism in the abelian category C. The image of f, denoted im f isthe subobject of B, ker(coker f). A sequence of maps in C

Af−→ B

g−→ C (7.1)

is called exact (at B) if kerg = im f. A sequence of maps is exact if it is exact at everyterm. A short exact sequence in C is an exact sequence of the form

0 → Af−→ B

g−→ C→ 0 (7.2)

A map of short exact sequences is composed of three maps φ,χ,ψ in C such that thefollowing diagram commutes

0 // A1g1//

φ

B1f1//

χ

C1//

ψ

0

0 // A2g2// B2

f2// C2

// 0

Clearly the short exact sequences in C form a category which we will denote as Sh(C).We similarly define a category LEHS(C) of long exact homology sequences, whichis the same as the category of exact positive chain complexes. Also we define thecategory LECS(C) of long exact cohomology sequences, which are the exact cochaincomplexes.

5FiXme: Define chain homotopy equivalence


7.4 Homology

7.8 Definition. If C is a chain complex, we define the subobjects of Cn: Zn(C) = kerdn (n-cycles),Bn(C) = imdn+1 (n-boundaries), and the subquotient Hn(C) = Zn(C)/Bn(C) (n-homology). For a cochain complex C, we have the subobjects of Cn: Zn(C) = kerdn

(n-cocycles), Bn(C) = imdn−1 (n-coboundaries), and the subquotient 6 Hn(C) =Zn(C)/Bn(C) (n-cohomology)

Observe that C is exact at Cn precisely when Hn(C) = 0. Also we have that Zn,Bn,Hndefine functors Ch(C) → C, and all together form functors Z∗,B∗,H∗ : Ch(C) → Gr(C).

7.9 Proposition. ([Wei94, page 7]) A sequence

0 → C ′ −→ C −→ C ′′ → 0 (7.3)

is exact in Ch(C) if and only if each sequence

0 → C ′n −→ Cn −→ C ′′n → 0 (7.4)

is exact in C.

7

7.5 Split Chain Complexes

7.10 Definition. We say that the chain complex C is split if there is a map s : C → C of degree 1 suchthat dsd = d.

7.11 Proposition. ([Web87, page 363]) Let C be a chain complex. The following conditions are equiva-lent:

1. C is split,

2. For every n, both dn : Cn → imdn is a split epimorphism and the inclusionimdn → Cn−1 is a split monomorphism.

7FiXme: put relationship with chain homotopy equivalence

7.5. Split Chain Complexes 71

3. For every n we can write

Cn ∼= imdn+1 ⊕Hn(C)⊕ imdn

so that dn becomes the map from imdn+1⊕Hn(C)⊕imdn to imdn⊕Hn−1(C)⊕imdn−1 sending (a,b, c) to (c, 0, 0).

Proof. To prove 2 implies 3, let sn be a splitting of dn and s ′n be a splitting of the inclu-sion. We have, using sn, that Cn ∼= kerdn ⊕ imdn. The map s ′n that gives the splitting ofthe inclusion can then be restricted to give a splitting of the inclusion imdn+1 → kerdn.Then, we have the isomorphism

Cn −→ imdn+1 ⊕Hn(C)⊕ imdnx 7−→ (s ′nx− s ′nsndnx, x− sndnx,dnx)

and that it has inverse

imdn+1 ⊕Hn(C)⊕ imdn −→ Cn(a,b, c) 7−→ a+ b− s ′n(b) + sn(c)

We can then check that dn has the desired rule of correspondence.

To prove 3 implies 1, define sn(x,y, z) = (z, 0, 0).

7.12 Corollary. Let C be a chain complex. The following conditions are equivalent:

1. C is split and exact.

2. C is chain homotopy equivalent to the zero complex, that is, C ' 0.

Proof. Suppose that C is split and exact, then by 3 from the previous proposition, we havethat Cn ∼= imdn+1⊕ imdn, and the expression for the boundary is (a, c) 7→ (c, 0). Definetn : imdn+1 ⊕ imdn → imdn+2 ⊕ imdn+1 by tn(a, c) = (0,a). It is immediate to checkthat dt+ td = 1. Now, if C ' 0, let t : Cn → Cn+1 such that dt+ td = 1. Composing onthe right with d, we get dtd = d, and so C is split. To check C is exact, take x ∈ Cn suchthat dnx = 0. Hence, applying tn−1dn + dn+1tn+1 = 1Cn

to it, we get dn+1tn+1x = x,that is, x is a boundary. Hence Hn(C) = 0.

7.13 Definition. If the chain complex C satisfies one of the conditions of the previous corollary, we sayit is contractible.


7.6 Exact Functors

7.14 Definition. Let F : C → D be an additive covariant functor between abelian categories. Then F is

called left exact if for any exact sequence of the form 0 → Af−→ B

g−→ C, we have thatthe following sequence is exact:

0 → FAFf−→ FB

Fg−→ FC. (7.5)

Now, F is right exact if for any exact sequence of the form Af−→ B

g−→ C → 0, thefollowing sequence is exact:

FAFf−→ FB

Fg−→ FC→ 0. (7.6)

The functor is exact if it is both left and right exact.A contravariant additive functor F : C → D is left exact (right exact, exact) if thecorresponding covariant functor Cop → D is left exact (right exact, exact). That is,

the contravariant F is left exact if for any exact sequence A f−→ Bg−→ C → 0 we have

that 0 → FAFf−→ FB

Fg−→ FC is exact, and it is right exact if for any exact sequence

0 → Af−→ B

g−→ C we have that FA Ff−→ FBFg−→ FC→ 0 is exact.

Let C be an abelian category andM be an object in C. Then we can consider the hom func-tor homC(M, −) as taking values in the category Ab. Similarly, we have a contravarianthom functor homC(−,M) : C → Ab. Both are clearly additive.

7.15 Theorem. ([Wei94, pages 27–28]) Let C be an abelian category. Then for any object M in C wehave that both homC(M, −), homC(−,M) : C → Ab are left exact.

8

7.16 Theorem. ([Wei94, page 28]) Let C be an abelian category. A sequence in C

Af−→ B

g−→ C (7.7)

is exact if the sequence

hom(M,A)f∗−→ hom(M,B)

g∗−→ hom(M,C) (7.8)

is exact for every M ∈ obj C.

8FiXme: GIVE PROOF

7.6. Exact Functors 73

7.17 Theorem. ([Wei94, page 51]) Let C and D be abelian categories, and R : C → D, L : D → C beadditive functors such that L a R. Then

1. L is right exact

2. R is left exact.

7.18 Definition. A non-empty small category I is called filtered if:

1. For every i, j ∈ obj I there is k ∈ obj I and morphisms i→ k, i→ j.

2. For every i, j ∈ obj I and every two morphisms u, v ∈ homI(i, j), there is k ∈ obj Iand a morphism w ∈ homI(j,k) such that wu = wv.

7.19 Theorem. ([Wei94, page 58]) If I is filtered, then both functors colim, lim : (R-mod)I → R-modare exact

9

9FiXme: NOW We could later include that Q IS FLAT, WHICH USES COR 2.6.17 IN WEIBEL.


8Appendix: Calculations

8.1 The Grothendieck Group

A semigroup is a set S together with a commutative and associative binary operation onS. A morphism f : S1 → S2 of abelian semigroups is a map f : S1 → S2 that preserves theoperation. We will use Abs to denote the category of abelian semigroups.

8.1 Definition. ([HS97, page 72]) Let S be an abelian semigroup. Then S×S has an induced structureof abelian semigroup. Define in S× S the relation (a,b) ∼ (c,d) if and only if there isu ∈ S such that a+d+u = b+c+u. This is an equivalence relation, and (S×S)/∼ is anabelian group, since [a,b]+[b,a] = [a+b,a+b] = [0, 0] = 0. This is the Grothendieckgroup of S, denoted Gr(S). Note that we have a homomorphism i : S → Gr(S). Weobserve the following universal property of i: If A is an abelian group and f : S → A

is a homomorphism, then there is f : Gr(S) → A making commute the followingdiagram

Si//

f

Gr(S)

f||z

zz

z

A

(8.1)

This universal property lets us verify that if E : Ab → Abs is the forgetful functor, wehave that Gr a E

75

76 Chapter 8. Appendix: Calculations

Examples 8.11. If S is the semigroup of all G-sets under disjoint union, then Gr(S) is the additive group of

what is called the Burnside ring of G and is denoted Ω(G),

2. If S is the semigroup of finitely generated kG-modules under direct sum, then Gr(S) is theadditive group of what is called the Green ring of G over k.

Bibliography

[BK72] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations.Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.

[HS97] P. J. Hilton and U. Stammbach. A course in homological algebra, volume 4 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition,1997.

[ML98] Saunders Mac Lane. Categories for the working mathematician, volume 5 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition,1998.

[Web87] Peter Webb. Subgroup complexes. In P. Fong, editor, The Arcata conferenceon representations of finite groups, volume 47 of Proc. of Symposia in PureMathematics, pages 349–365, 1987.

[Wei94] Charles A. Weibel. An Introduction to Homological Algebra. Cambridge Uni-versity Press, 1994.

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