For the rest of this chapter we study adjunctions. In particular, we show that adjunctions are unique up to unique isomorphism, and we give a criterion to prove that two functors are adjunctions, which means that one functor is left adjoint to other functor and the other is right adjoint to the first functor. We follow [Bor94a, 3.1]. Let us start by a definition of adjoints.
Definition 1.3.1 (Adjoint). A functorF :CÑDis left adjoint to a functor G:DÑC if there exists a natural transformationη: IdC ÑGF such that for all objectsC ofC the pairpFpCq, ηpCqqis the universal object fromC to G. This means that for any morphism f :CÑGpDqin C there exists a unique morphismh:FpCq ÑD in D such that for allDPObD the pairpGpDq, pDqqis the universal object fromF to D. That is, for any morphism f : FpCq Ñ D in D there exists a unique morphism h : C Ñ GpDq in C such that the following diagram is
The following lemma shows that the left and right adjoints are unique up to unique isomorphism.
Lemma 1.3.2. Let F :CÑD be a functor and G, G1:DÑC right adjoints to F with φ, φ1 the isomorphisms on the set of morphisms. Then G–G1. Similarly, if F, F1:CÑD are left adjoints to G, thenF –F1.
Proof. Let:F GÑIdD and1:F G1ÑIdD be natural transformations such that for any objectDPDthe pairs pGpDq, pDqqand pG1pDq, 1pDqq are universal objects fromF toD. By definition of universal object, there exist unique morphismsh:GpDq ÑG1pDqand h1 :G1pDq ÑGpDqsuch thathh1 “IdG1pDq and h1h“IdGpDq. These morphisms show thatG–G1.
Similarly, let η : IdC ÑGF and η1 : IdC ÑGF1 be natural transformations such that for any object C of C pFpCq, ηCq and pF1pCq, η1pCqq are universal objects from C to G. By definition there exists unique morphisms h:FpCq ÑF1pCqandh1:F1pCq ÑFpCqsuch thathh1“IdFpCq andh1h“IdF1pCq. These morphisms show that the functorsF andF1 are isomorphic.
The following lemma gives a relation of left and right adjoint of categories and their opposite categories. We denote by fop, Fop, and τop the natural morphisms, functors, and natural transformations, respectively, in the corresponding opposite categories. Note thatpfopqop“f,pFopqop“F, andpτopqop“τ.
Lemma 1.3.3. A functor F :C ÑDis the left adjoint of G:DÑC if and only if Fop:Cop ÑDop is the right adjoint ofGop:DopÑCop.
Proof. ñ: Suppose thatF is the left adjoint ofG. Letη : IdC ÑGF be the natural transformation such that for any objectCofCthe pairpFpCq, ηpCqqis the universal object fromCtoG. Thus for any morphismf :CÑGpDq there exists a unique morphism h: GFpCq ÑGpDqsuch that the following diagram is commutative in the dual category
This shows thatpFoppCq, ηoppCqqis the universal object fromGop toC. HenceFopis the right adjoint ofGop.
This shows that the pairFpCq, pCqqis a universal object fromC toG. ThereforeF is the left adjoint toG.
We will use later the following theorem in chapter 7 to the theory oft-structures to prove that abstract truncations are adjoints to inclusion functors.
Theorem 1.3.4. LetF :CÑDandG:DÑC be functors. Then the following conditions are equivalent.
(i) F is left adjoint to G.
Proof. piq ñ piiq : The natural transformation η : IdC Ñ GF is given by the definition of left adjoint, so let us construct the natural transformation . Consider the universal object pF GpDq, ηpGpDqqq from GpDq to G. Let pDq : F GpDq Ñ D be the unique morphism given by the definition of universal object such that the following diagram is commutative
To show thatis a natural transformation, let d:DÑD1 be a morphism inD. Then
GppD1q ˝F Gpdqq ˝ηpGpDqq “GppD1qq ˝GF Gpdq ˝ηpGpDqq “GppD1qq ˝ηpGpD1qq ˝Gpdq “Gpdq, and
Gpd˝pDqq ˝ηpGpDqq “Gpdq ˝GppDqq ˝ηpGpDqq “Dpdq.
By uniqueness of the factorization of the universal objectpD1q ˝F Gpdq “d˝pDq. This shows thatis a natural transformation.
To show the commutativity of the second triangle of (1.3), letCPObC andpGFpCq, ηpCqqthe universal object fromFpCqto F. Then
GppFpCqq ˝F ηpCqq ˝ηpCq “GpFpCqq ˝GF ηpCq ˝ηpCq
“ηpCq “GpIdFpCqq ˝ηpCq.
By uniquness of the factorization of universal object, pGpDqq ˝GηpDq “ IdFpCq. Hence the second triangle is commutative.
piiq ñ piiiq : Given a morphism d : FpCq Ñ D, we define φC,Dpdq to be the composite Gpdq ˝ηpCq. For a morphismc:CÑGpDqwe defineτC,Dpcqto be the compositepDq ˝Fpcq. We have
pτC,D˝φC,Dqpdq “τC,DpGpdq ˝ηpCqq “pD1q ˝FpGpdq ˝ηpCqq “pD1q ˝F Gpdq ˝FpηpCqq
“d˝pFpCqq ˝FpηpCqq “d, and
pφC,D˝τC,Dqpcq “φC,DppDq ˝Fpcqq “GppDq ˝Fpcqq ˝ηpCq “GppDqq ˝GFpcq ˝ηpCq
“GppDqq ˝ηpC1q ˝c“c,
so the mapsτC,D andφC,D are mutual inverses. This shows thatφC,D is bijective for all CPObC andDPObD. To show that the diagram (1.4) is commutative, let f : C1 Ñ C and g : D Ñ D1 be morphisms in C and D, respectively. Letd:FpCq ÑDPMorDpFpCq, Dq. Then
ppGpgq ˝ ´ ˝fq ˝φC,Dqpdq “ pGpgq ˝ ´ ˝fqpGpdq ˝ηpCqq “Gpgq ˝Gpdq ˝ηpCq ˝f
“Gpgq ˝Gpdq ˝GFpfq ˝ηpC1q, and
pφC1,D1˝ pg˝ ´ ˝Fpfqqqpdq “φD1,C1pg˝d˝Fpfqq “Gpg˝d˝Fpfqq ˝ηpC1q
“Gpgq ˝Gpdq ˝GFpfq ˝ηpC1q.
This shows the commutativity of the diagram.
piiiq ñ piq: First note that for any objectCPObC the morphismφC,FpCqpIdCq:CÑGFpCqdefines a natural transformation by commutativity of the diagram (1.4). It suffices to show that for any object C of C the pair pFpCq, φC,CpIdCqqis a universal object fromCtoG. Letf :CÑGpDqbe a morphism inDand letg:FpCq ÑD be the morphismτC,Dpfq. Now
ppGpgq ˝ ´ ˝IdCq ˝φC,FpCqqpIdFpCqq “ pGpgq ˝ ´ ˝IdCq ˝φC,FpCqpIdFpCqq “ pGpgq ˝φC,FpCqqpIdFpCqq
“ pφC,D˝gqpIdFpCqq “ pφC,D˝ pg˝ ´ ˝IdFpCqqqpIdFpCqq,
so the pair is the universal object. To show uniqueness of g, let g1 : FpCq Ñ D be a morphism such that pGpg1q ˝φC,FpCqqpIdFpCqq “f. Then
φC,Dpg1q “φC,Dppg1˝ ´ ˝IdFpCqqpIdFpCqqq
“ pGpg1q ˝ ´ ˝IdCqpφC,FpCqpIdFpCqq
“Gpg1q ˝φC,FpCqpIdFpCqq
“f “φC,Dpgq,
where the second equality follows from commutativity of the diagram (1.4). SinceφC,Dis bijection, this shows that g“g1.
pivq ô piiiq: Suppose Gis the right adjoint ofF. By lemma 1.3.3Gop is the left adjoint ofFop. By (iii) the following diagram is commutative for all morphismsgop:D1ÑD andfop:CÑC1
DoppGoppDq, Cq CoppD, FoppCqq
DoppGoppD1q, C1q CoppD1, FoppC1qq
φD,C
f˝´˝Goppgopq Foppfopq˝´˝gop φD1,C1
Taking the dual we obtain the diagram (1.4).
Conversely, suppose that (iii) holds. For any morphismsg:DÑD1 andf :C1ÑCconsider the corresponding diagram of (1.4) in the dual category
MorDoppGoppDq, Cq MorCoppD, FoppCqq
MorDoppGoppD1q, C1q MorCoppD1, FoppC1qq
φopD,C
Goppgopq˝´˝fop gop˝´˝Foppfopq φop
D1,C1
Sincepiiiq ñ piq,Gop is the left adjoint ofFop. By lemma 1.3.3Gis the right adjoint ofF. The following example is basic adjunction in commutative algebra.
Example 1.3.5(Adjunction inRMod). LetRbe a commutative ring. One can show that in the categoryRMod the the functor N b ´ :RMod Ñ RModis left adjoint to the functor MorRp´, Nq :RMod ÑRMod. This means that for allR-modulesM andP the bijections
MorRModpM bRN, Pq –MorRModpM,MorRModpP, Nqq
are natural in bothM andP so that any diagram of the form (1.4) commutes. Actually, since the categoryRMod isR-linear, meaning that all sets of morphisms admit a naturalR-module structure, these bijections between sets of morphisms are isomorphisms ofR-modules. For more details, see [Bor94a, Example 3.1.6.e].
Chapter 2
Abelian categories
In this chapter we introduce the main mathematical objects to study in this thesis, additive and abelian categories.
Abelian categories can be seen as categorical generalization of the categories of R-modules, because every small abelian category admits a full, faithful, and exact embedding to RMod, for some commutative ringR [Bor94b, Theorem 1.14.9]. Thus one can use intuition from the category of R-modules to study abelian categories. Every abelian category is additive, by theorem 2.3.3, but not every additive category is abelian.
2.1 Additive categories
Let us start with the definitions of preadditive and additive categories.
Definition 2.1.1 (Preadditive and additive categories). Apreadditive category Ais a category such that for any X, Y PObAthe set MorApX, Yqhas a structure of an abelian group, and for any morphismsf1, f2:X ÑY and g1, g2:Y ÑZ in Awe have
pg1`g2qpf1`f2q “g1f1`g1f2`g2f1`g2f2.
We say that a preadditive categoryAisadditive if it has a zero object, denoted by 0, and biproducts, that is, for anyX, Y PObAthere exists an objectX‘Y and morphismsi1:X ÑX‘Y,p1:X‘Y ÑX,i2:Y ÑX‘Y, andp2:X‘Y ÑY such that the following equalities hold
p1i1“IdX, p2i2“IdY, p1i2“0, p2i1“0, i1p1`i2p2“IdX‘Y,
and for any objectZand any morphismsf :X ÑZandg:Y ÑZthere exist a unique morphismf‘g:X‘Y ÑZ such that the following diagram is commutative
Z
X X‘Y Y
f i1
f‘g
p1 p2
g i2
One can easily verify that the morphismf‘g is given byf p1`gp2.
Example 2.1.2 (Preadditive and additive categories). Let us give a few examples concerning preadditive and additive categories.
(i) The category of groups is not preadditive. The reason is that there is no natural way to define abelian category structure on the set of morphisms. For details, see [Bor94b, Example 1.2.9.b].
(ii) Consider the full subcategoryCofCRingconsisting of the objectZ. Clearly MorCpZ,Zq “Zand this set has a natural structure of an abelian group given by sum of ring homomorphisms. For any ring homomorphisms f1, f2, g1, g2:ZÑZone has
pf1`f2qpg1`g2q “f1g1`f1g2`f2g1`f2g2. Hence the category C is preadditive.
On the other hand, this category is not additive. Indeed, it does not have the zero object because MorCpZ,Zq consists of more than one morphism. Also, C does not have biproducts. Indeed, supposepZ, i1, i2, p1, p2q is the biproduct of Zand Z. From the identities p1i1 “IdZ and p2i2 “IdZ it follows thati1 and i2 send the identity element 1 of Z to ´1 or 1, because multiplications by 1 and ´1 are the only automorphisms of Z. Let 2 : ZÑZand 3 : ZÑZbe the multiplications by 2 and 3. Then there cannot be a morphism hsuch that the following diagram would commute
Z
Z i1 Z Z
2 h
i2 3
because h would need to map the element 1 to (2 or ´2) and (3 or ´3). Hence the category C is only preadditive.
(iii) Let us show that it is not enough that a preadditive category to have a zero object to be additive. Consider the full subcategryC ofCRingconsisting of the objects 0 andZ. Clearly 0 cannot be the biproduct ofZand Z, so the argument of the previous example (ii) shows that the the biproduct ofZandZdoes not exist inC. (iv) For only this example, to keep the argument readable, we abuse introduced notation and writein andpn for the inclusion and projection of thenth component of biproduct and not care in which order the biproduct is formed.
In this example we show that there exists only one finite additive category, up to equivalence of categories, the category having only the zero object. Clearly the category having only the zero object is additive. Suppose that Ais an additive category, and let X be a nonzero object ofA. Then MorApX, Xq ě2 because this set must contain at least the zero morphism and the identity morphism. Here it cannot be the case that IdX
equals the zero morphism because otherwiseX would be isomorphic to zero object and hence a zero object.
This contradicts the assumption that X is not a zero object.
Letně2 and letf :‘ni“1X Ñ ‘ni“1X be a nonzero morphism. This means that f ij‰0 for some 1ďjďn, because f “fpi1p1`. . .`inpnq. The morphismsfpi1p1`. . .`inpnq, fpi1p1`. . .`ijˆpj`. . .`in`1pn`1q:
‘n`11 X Ñ ‘n1X are different. Indeed, we have
fpi1p1`. . .`inpnqij“f ij ‰0“fpi1p1`. . .`ijˆpj`. . .`in`1pn`1qij. Hence
#pMorAp‘n`11 X,‘n1Xqq ě2¨#pMorAp‘n1X,‘n1Xqq ´1.
For any two distinct morphisms φ, φ1 :‘n`11 X Ñ ‘n1X, we have pjφ‰ pjφ1, for some 1ď j ďn, and the positive integernăm. By induction, the inequality (2.1) now says
#pMorAp‘mi“1X,‘mi“1Xqq ą#pMorAp‘ni“1X,‘ni“1Xqq.
This is a contradiction. Thus a finite additive category Acannot contain a nonzero object.
The following proposition shows that in a preadditive category, the biproduct of two objects is both the product and the coproduct of the objects.
Proposition 2.1.3. LetAbe a preadditive category and let AandB be objects ofA. ThenAhas the biproduct of A andB if and only if it has the product and the coproduct ofA andB.
A functor which preserves the structure of an additive category is called an additive functor.
Definition 2.1.4 (Additive functor). LetF :AÑB be a functor between additive categories. If for all objects A1, A2of Athe map
MorApA1, A2q ÑMorBpFpA1q, FpA2qq, f ÞÑFpfq is a group homomorphism, thenF isadditive.
The following proposition gives a characterization of an additive functor by biproducts.
Proposition 2.1.5. A functorF :AÑBbetween additive categories is additive if and only if preserves biproducts, that is, ifpA‘B, i1, i2, p1, p2qis a biproduct in A, thenpFpA‘Bq, Fpi1q, Fpi2q, Fpp1q, Fpp2qqis a biproduct inB. In particular, an additive functor preserves the zero object.
Proof. ñ: LetF be additive and letpA‘B, i1, i2, p1, p2qbe a biproduct inA. Now, Fpp1i1q “IdFpAq, Fpp2i1q “ IdFpbq, Fpp1i2q “0,Fpp2i1q “0, and IdFpA‘Bq“Fpi1p1`i2p2q “Fpi1qFpp1q `Fpi2qFpp2q. Indeed, to see that Fpp1i2q “Fpi2p1q “0 it suffices to show thatFp0qis the zero object inB. Since
MorAp0A,0Aq ÑMorBpFp0Aq, Fp0Aqq
is a group homomorphism,FpId0qfactors through the zero object ofB. Thus there exist morphismsf :Fp0q Ñ0 andg: 0ÑFp0q, which are unique, such thatgf “IdFp0q. Nowgf “Id0by uniqueness of the morphism from the zero object to itself. HenceFp0qis isomorphic to the zero object inB and is itself the zero object.
To see that pFpA‘Bq, Fpi1q, Fpi2q, Fpp1q, Fpp2qqis the biproduct of FpAqand FpBqin B it suffices to show that it is isomorphic to the object of the biproductpFpAq ‘FpBq, i1, i2, p1, p2q. By the definition of biproduct there exists a unique morphismh:FpA‘Bq ÑFpAq ‘FpBqsuch that p1h“Fpp1qandp2h“Fpp2q. Now
IdFpA‘Bq“Fpi1qFpp1q `Fpi2qFpp2q “Fpi1qp1h`Fpi2qp2h“ pFpi1qp1`Fpi2qp2qh and
IdFpAq‘FpBq“i1p1`i2p2“i1IdFpAqp1`i2IdFpBqp2“i1Fpp1qFpi1qp1`i2Fpp2qFpi2qp2
“i1p1hFpi1qp1`i2p2hFpi2qp2“hFpi1qp1`hFpi2qp2“hpFpi1qp1`Fpi2qp2q.
This shows that his an isomorphism and thatFpA‘Bqis the biproduct ofFpAqandFpBq. Hence F preserves biproducts.
ð: First we show that F preserves the zero object. By assumption pFp0‘0q, Fpi1q, Fpi2q, Fpp1q, Fpp2qq is the biproduct of Fp0q and Fp0q in B. Since 0 is the zero object in A, Fpi1q “ Fpi2q and Fpp1q “ Fpp2q. Let f1, f2:BÑFp0qand g1, g2:Fp0q ÑC be morphisms inB, and leth1 :B ÑFp0‘0qandh2:Fp0‘0q ÑC be the unique morphisms such thatFpp1qh1“f1,Fpp2qh1“f2,h2Fpi1q “g1, andh2Fpi2q “g2. Now
f1´f2“Fpp1qh1´Fpp2qh1“ pFpp1q ´Fpp2qqh1“0 and
g1´g2“h2Fpi1q ´h2Fpi2q “h2pFpi1q ´Fpi2qq “0.
Hencef1“f2andg1“g2. This shows thatF preserves the zero object.
It remains to show that F preserves difference of two morphisms. Letf, g:A1ÑA2be morphisms inA. From Fpf´gq “Fppp1´p2qpi1f`i2gqq “Fpp1´p2qFpi1f`i2gq “Fpp1´p2qpFpi1qFpfq `Fpi2qFpgqq, we see that it is enough to show thatF preserves the differencep1´p2. Now
Fpp1´p2qFpi1q “Fppp1´p2qi1q “FpIdA1q “IdFpA1q“ pFpp1q ´Fpp2qqFpi1q “IdA2
and
Fpp1´p2qFpi2q “Fppp1´p2qi2q “FpIdA2q “IdFpA2q“ pFpp1q ´Fpp2qqFpi2q “ ´IdA2, so by the universal property of the biproductFpp1´p2q “Fpp1q ´Fpp2q. This completes the proof.
Example 2.1.6 (Additive representable functor). Let us give an example how to construct additive functors on an additive category. Suppose we have an additive category A. Then for any object A P ObA we can consider the representable functor MorApA,´q, item (ii). Let us show that this is an additive functor. Fix two morphisms f, g:X ÑY. Then for any morphismh:AÑX we have
MorApA,´qpf´gqphq “ pf´gqphq “MorApA,´qpfqphq ´MorApA,´qpgqphq.
This shows that the functor MorApA,´q is additive.
We conclude this section by an additive version of Yoneda’s lemma 1.1.7. This is a special case of a more general version, the enriched Yoneda’s lemma [Bor94b, Theorem 6.3.5].
Proposition 2.1.7 (Additive Yoneda’s lemma). Let A be an additive category and F : A Ñ Ab an additive functor. For any objectA ofA we have an isomorphism of abelian groups
θF,A:N atpMorApA,´q, Fq ÑFpAq,
which is naural in both A and F. This means that for any morphism φ : A Ñ A1 in A and for any natural transformationη:F ÑGthe following diagrams are commutative
N atpMorApA,´q, Fq FpAq
N atpMorApA1,´q, Fq FpA1q
θF,A
´˝p´˝φq Fpφq
θF,A1
N atpMorApA,´q, Fq FpAq
N atpMorApA,´q, Gq GpAq
θF,A
η˝´ ηpAq
θG,A
.
Proof. Let α : ApA,´q Ñ F be a natural transformation and define θF,Apαq “ αpAqpIdAq. Conversely, for any elementaPFpAqwe assign a natural transformation
τpaqpBq: MorApA, Bq ÑFpBq, τpaqpBqpfq “Fpfqpaq.
Here the morphism τpaqpBq is a group homomorphism because F is an additive functor. By the proof of theo-rem 1.1.7 this is map is bijective, hence isomorphism of abelian groups.
To see that this map is natural in A and F, let φ : A Ñ A1 be a morphism in A, η : F Ñ G a natural transformation, and let ΨPN atpMorpA,´q, Fq. Then
pFpφq ˝θF,AqpΨq “FpφqpΨpAqpIdAqq “ΨpA1qpφq
“θF,A1pΨ˝ p´ ˝φqq “ pθF,A1˝ p´ ˝φqqpΨq,
where the second equality follows from the fact that Ψ is a natural transformation, so the first diagram is commu-tative. Also
pηpAq ˝θF,AqpΨq “ηpAqpΨpAqpIdAqq “θG,Apη˝Ψq
“ pθG,A˝ pη˝ ´qqpΨq, so the second diagram is also commutative.