Let us begin with the definition of a category.
Definition 1.1.1 (Category). A category C consists of a class of objects ObC, a class of morphisms MorC which associates to every pair of elements X, Y P ObC a set MorCpX, Yq, also denoted by CpX, Yq, and three maps d, c: MorCÑMorC, and˝ called domain, codomain, and composition, such that the following conditions hold
C 1 For anyf PMorCpX, Yq, we definecand dby
dpfq “X and cpfq “Y.
The composition˝ is a map defined from the class ď
X,Y,ZPObC
MorCpY, Zq ˆMorCpX, Yq
to MorC, such that for any morphismsf PMorCpX, YqandgPMorCpY, Zqthe image ofpg, fqis contained in MorCpX, Zq. For any two morphisms f P MorCpX, Yq and g P MorCpY, Zq we write g˝f, orgf, for
˝pg, fq.
C 2 For anyX, Y, Z, W PObC, f PMorCpX, Yq,g PMorCpY, Zq, andhPMorCpZ, Wqthe composition map satisfies
h˝ pg˝fq “ ph˝gq ˝f.
For this reason we usually omit brackets for composition of morphisms.
C 3 For any object X P ObC there exist a morphism IdX P MorCpX, Xq such that for any Y P ObC and f PMorCpX, Yqwe have
f˝IdX “f and IdY˝f “f.
If ObC is a set, then the categoryC is said to besmall.
The elements of ObCare called the objects of the categoryCand the elements of MorCare called the morphisms ofC. One can easily verify that for any category C theopposite category Cop, obtained by defining ObCop “ObC and MorCoppX, Yq:“MorCpY, Xqfor all objectsX andY, is a category.
Let C and D be categories. We say that C is a subcategory of Dif ObC ĂObD and for anyX, Y PObC we have MorCpX, Yq ĂMorDpX, Yq.
Example 1.1.2. Here are some examples of categories. We leave it for the reader to verify that these are categories.
(i) The category of setsSetconsists of all the sets as objects and for any setsX andY we let MorSetpX, Yqto consist of all functions fromX toY.
(ii) The category of abelian groupsAbconsists of all abelian groups and for any abelian groupsAandB the set MorAbpA, Bqconsists of all the group homomorphisms fromAto B.
(iii) The category of commutative rings CRing is the category where the objects are commutative rings and MorCRingpR, Sqis the set of all ring homomorphisms fromR toS for any commutative ringsR andS.
(iv) Fix a commutative ring R. The category of R-modules RMod consists of all R-modules and the set MorRModpM, Nqconsists of allR-module homomorphisms from M toN for anyR-modulesM andN.
(v) The category of topological spacesTopconsists of topological spaces and continuous maps between them.
(vi) Fix a topological space X. Denote by ToppXq the category where objects are the open subsets ofX and morphisms are inclusions, that is, ifU andV are open subsets such thatU ĂV, then the set MorToppXqpU, Vq consists of one element and otherwise MorToppXqpU, Vq is an empty set. It is easy to see that this is a subcategory of Top.
For any mathematical objects one usually wants to consider maps which preserve the structure of the object.
For categories such a map is called a functor.
Definition 1.1.3 (Functor). A functor F : C Ñ D from a category C to a category D consists of two maps ObC ÑObDand MorCÑMorD, both denoted byF, such that for any objectX PObC,FpIdXq “IdFpXq and for any two morphismsf, gPMorC, such thatgf is defined, we haveFpgfq “FpgqFpfq. One can easily check that composition of two functors is a functor.
LetF :C ÑDbe a functor. If for any objectsX, Y PObC the mapF : MorCpX, Yq ÑMorDpFpXq, FpYqqis injective (resp. surjective, resp. bijective), thenF is calledfaithful (resp. full, resp. fully faithful). A subcategory C ofDis calledfull if the inclusion functor is full.
For a categoryC we denote by IdC the functor which is an identity both on objects and morphisms.
Example 1.1.4. Here are some examples of functors. The verification of these being functors is left to the reader.
(i) The category of small categoriesCat consists of all small categories and the set of morphisms between two small categories consists of all functors between the categories.
(ii) Let C be a category. For any object X P ObC we define a functor MorCpX,´q : C Ñ Set, called the representable functor ofX, which maps an objectY ofC to MorCpX, Yq. Iff :Y ÑZ is a morphism inC, then MorCpX,´qpfq: MorCpX, Yq ÑMorCpX, Zqis given by composition with f, i.e., ψÞÑf ˝ψ. One can check that this defines a functor.
An object X in a categoryC is initial if for any object Y ofC there exists a unique morphism from X to Y. If for any objectY there exists a unique morphism fromY to X, then the objectX is terminal. An object both initial and terminal is azero object. If a category has a zero object, we call the composite X Ñ0 ÑY thezero morphism fromX toY.
A morphism f : X ÑY in a category C is a monomorphism if for any two morphisms g1, g2 : Z Ñ X such thatf g1“f g2 we haveg1“g2. The morphismf is anepimorphism if for any two morphismg1, g2:Y ÑZ with g1f “ g2f we have g1 “g2. A morphism f : X ÑY is an isomorphism if there exists a morphism g : Y ÑX such that gf “ IdX and f g “IdY. One can easily show that an isomorphism is both a monomorphism and an epimorphism.
Functors can be viewed as morphisms of categories. To understand morphisms of categories better we define morphisms of morphisms of categories which can be thought of as some kind of homotopies between morphisms. In category theory such morphisms are called natural transformations.
Definition 1.1.5(Natural transformation). Anatural transformation τ :F ÑGof functorsF, G:CÑDconsists of a morphismτpDq:FpDq ÑGpDqfor any objectDofDsuch that for any morphismf :X ÑY ofCthe diagram
FpXq GpXq
FpYq GpYq
τpXq
Fpfq Gpfq
τpYq
commutes.
LetF, G:CÑDbe functors. We say that the functorsF andGareisomorphic, writtenF –G, if there exists a natural transformationτ :F ÑGsuch that for every object X PC the morphismτpXq:FpXq ÑGpXq is an isomorphism inD.
Example 1.1.6. Here are some examples of natural transformations.
(i) LetCandDbe objects in a categoryCand letf :CÑDbe a morphism inC. Then the morphismf induces a natural transformation between the representable functors MorCpD,´qand MorCpC,´q, see example 1.1.4 (ii), denoted by ´ ˝f, which maps a morphism φ: D Ñ X to φ˝f : C ÑX. Indeed, for any morphism g:XÑY we have
pMorCpC,´qpgq ˝ p´ ˝fqpXqqpφq “g˝φ˝f “ pp´ ˝fqpYq ˝MorCpD,´qpgqqpφq, so ´ ˝f is a natural transformation.
(ii) For any two categoriesC andD, we can define the category of functorsFunpC,Dq, also denoted CD, fromC to D. Morphisms in this category are natural transformations of functors.
(iii) LetX be a topological space andToppXqthe category defined in example 1.1.2 (vi). The category FunpToppXqop,Setq(resp. FunpToppXqop,Abq, resp. FunpToppXqop,CRingq, resp. FunpToppXqop, RModq) is called the category of presheaves of sets (resp. abelian groups, resp. commutative rings, resp.
R-modules) onX.
Next we prove a well-known result which identifies natural transformations from a representable functor to a functorF with a set defined byF.
Theorem 1.1.7 (Yoneda’s lemma). Let C be a category, X an object of C, and MorCpX,´q the representable functor (ii). For any functor F :CÑSetwe have a bijection
θF,X :N atpCpX,´q, Fq ÑFpXq,
whereN atpCpX,´q, Fqdenotes the class of natural transformations fromCpX,´qtoF. Proof. For any natural transformationσ:CpX,´q ÑF, defineθF,Xpσq “σpXqpIdXq.
For any xPFpXqwe define a natural transformationτpxq:CpX,´q ÑF as follows. For any object Y PC, let τpxqpYq:CpX, Yq ÑFpYq, be the map f ÞÑFpfqpxq. Then for any morphism g :Y ÑZ PMorC the following diagram
CpX, Yq FpYq
CpX, Zq FpZq
τpxqpYq
g˝ Fpgq
τpxqpZq
commutes. This shows that τpxqis a natural transformation.
It suffices to show thatτpxqis the inverse of θF,X. For anyxPFpXqwe have θF,Xpτpxqq “τpxqpXqpIdXq “FpIdXqpxq “x and
τpθF,XpσqqpYqpfq “τpσpXqpIdXqqpYqpfq “ pFpfqσpXqqpIdXq “σpYqpfq, where the last equality follows from the following commutative diagram
CpX, Xq FpXq
CpX, Yq FpYq
σpXq
f˝ Fpfq
σpYq
applied to IdX. HenceθF,X is bijective.
By using Yoneda’s lemma we can easily identify isomorphic representable functors.
Corollary 1.1.8. Let C be a category and X, Y P ObC. Then F : MorCpX,´q Ñ MorCpY,´q, given by some morphism f :Y ÑX, by theorem 1.1.7, is an isomorphism if and only if f is an isomorphism.
Proof. ñ: Suppose that F is an isomorphism. Let g “ pFpYqq´1pIdYq : X Ñ Y be a morphism in C. Then g˝f “IdY. Let G: MorCpY,´q ÑMorCpX,´q be the natural transformation induced by the morphismg like in theorem 1.1.7. Now, let φ “ pGpXqq´1pIdXq: Y Ñ X. We have φ˝g “IdX. Now φ“ φ˝IdY “φ˝g˝f “ IdX˝f “f. Hencef˝g“IdX andf is an isomorphism.
ð: Suppose thatf :Y ÑX is an isomorphism. Then for any objectZ ofC, the mapFpZqis injective because for any morphism h:X ÑZ, h˝f˝f´1“h. To see that FpZqis surjective, let ψ:Y ÑZ be any morphism of C. Then the morphismψf´1is mapped toψ byFpZq. This completes the proof.
Definition 1.1.9 (Equivalence). A functor F : C Ñ D is an equivalence of categories if there exists a functor G:DÑC such thatGF –IdC andF G–IdC.
A functor F :CÑDis said to beessentially surjective if for any objectX ofD there exists some objectY of C such thatFpYq –X.
Theorem 1.1.10(Criteria for equivalence). A functorF :CÑDis an equivalence if and only if it is fully faithful and essentially surjective.
Proof. ñ: Suppose F is an equivalence of categories. By definition there exist a functor G : D Ñ C such that τ1:GF –IdC andτ2:F G–IdD. For any objectXPD, the morphismτ2pXq:F GpXq ÑX is an isomorphisms, faithful, we obtaing“Fpfq. This shows thatF is full.
ð: Let F be fully faithful and essentially surjective. We define a functorG:DÑC as follows: for any object Y ofDfix an isomorphismY :Y ÑF X and an objectX ofC. DefineGpYq “X. For any morphism g:Y1ÑY2
It remains to show thatF G–IdD and GF –IdC. Commutativity of the unique morphism such thatFpηpXqq “´1FpXq. SinceF is fully faithful, commutativity of
GFpX1q X1
Commutativity of the latter diagram follows from commutativity of (1.1). This shows thatGF –IdC.
Example 1.1.11 (Example of equivalence). The above theorem shows that the inclusion functor of a full subcat-egoryDof a categoryCis an equivalence of categories if and only if every object of Cis isomorphic to some object ofD.