In this section we follow [Bor94b, Section 1.6] to prove that an abelian category is additive. This result is inter-esting in the sense that in the definition of abelian category 2.2.1 we didn’t touch the set of morphisms between objects directly. Nevertheless, this definition gives the sets of morphisms between objects the structure of abelian groups, that is, MorpX, Yqis an abelian group for any objectsX andY, which are compatible with composition of morphisms. This means that for any morphismsf, g:X ÑY andh, k:Y ÑZ we havehpf `gq “hf`hg and ph`kqf “hf`kf.
Lemma 2.3.1. Let Abe an abelian category. For any objectAofAthe cokernelq:AˆAÑQof∆ :AÑAˆA, the unique morphism induced by the morphisms IdA andIdA, has the property thatQ–A.
Proof. Denote by r : A Ñ Q the composite qpIdA,0q. We prove that r is an isomorphism. Fix notation by the following commutative diagram
We have ∆ “ kerpcokerp∆qq “ kerpqq by corollary 2.2.11 because ∆ is a monomorphism. From p1pIdA,0q “ IdA “ p2p0,IdAqit follows that p1 and p2 are epimorphisms and pIdA,0q and p0,IdAq are monomorphisms. Let
the universal property gives p0,IdAqp2v “ v. The factorization p2v : V Ñ A is unique because p0,IdAq is a thusx“0, becauseq is an epimorphism. Thereforer is an epimorphism, by 2.2.6, and we conclude that r is an isomorphism.
Let us introduce the sum of morphisms for an abelian category A. Denote by q the cokernel of the diagonal morphism ∆ :AÑAˆAand writeσAfor the compositer´1q, whereris the compositeqpIdA,0qas in lemma 2.3.1.
The composite
B pf,gq AˆA σA A
is denoted byf´g. We definef`g“f´ p0´gqand show that this definition makes an abelian category additive.
To do that, we need the following lemma.
Lemma 2.3.2. Let A be an abelian category and keep the notation above. For a morphismf : B ÑA in A, we havef ˝σB“σA˝ pf ˆfq.
Proof. Fix notation by the following commutative diagram
B A
the universal property for product impliespfˆfqpIdB,0q “ pIdA,0qf. Putting these together we get g“gσBpIdB,0q “σApfˆfqpIdB0q “σApIdA,0qf “f.
We are ready to prove that an abelian category is additive.
Theorem 2.3.3. An abelian categoryAis additive.
Proof. LetA, B, C be any objects inA, andf :CÑB a morphism inA.
MorApA, Bq abelian: Let a, b, c, d : C Ñ A be any morphisms in A. By lemma 2.3.2 for p1 : AˆA Ñ A and p2:AˆAÑAwe have the following commutative diagrams
pAˆAq ˆ pAˆAq AˆA
AˆA A
p1ˆp1
σAˆA σA
p1
pAˆAq ˆ pAˆAq AˆA
AˆA A
p2ˆp2
σAˆA σA
p2
Now, by commutativity
#p1ppa, bq ´ pc, dqq “p1σAˆAppa, bq,pc, dqq “σApp1ˆp1qppa, bq,pc, dqq “σApa, cq p2ppa, bq ´ pc, dqq “p2ppa, bq,pc, dqqσAˆA“σApp1ˆp1qppa, bq,pc, dqq “σApb, dq Since
#p1ppa´cq,pb´dqq “a´c“σApa, cq p2ppa´cq,pb´dqq “b´d“σApb, dq by uniqueness we have
pa, bq ´ pc, dq “ pa´c, b´dq.
Lemma 2.3.2 applied toσA:AˆAÑA, we get the following commutative diagram pAˆAq ˆ pAˆAq AˆA
AˆA A
σAˆσA
σAˆA σA
σA
Now
pa´cq ´ pb´dq “σAppa´cq,pb´dqq
“σAppa, bq ´ pc, dqq
“σAσAˆAppa, bq,pc, dqq
“σApσAˆσAqppa, bq,pc, dqq
“σApσApa, bq, σApc, dqq
“σAppa´bq,pc´dqq
“ pa´bq ´ pc´dq,
where the equality pσAˆσAqppa, bq,pc, dqq “ pσApa, bq, σApc, dqqfollows from the universal property of the product. Indeed, suppose we have a morphismφ:Y ÑZ and morphismsx, y:X ÑY. Then
#p1pφx, φyq “φx p2pφx, φyq “φy
and from commutativity of the following diagram
X Y Z
XˆX Y ˆY ZˆZ
X Y Z
x φ
px,yq p1
p2
φˆφ p1
p2
p1
p2
y φ
we get that
#p1pφˆφqpx, yq “φx p2pφˆφqpx, yq “φy.
Hence pφˆφqpx, yq “ pφx, φyq. DenotingB byx,AˆAbyY and Z,σA byφ, and aandb byxand y one gets the claimed equality.
We are ready to verify the axioms of an abelian group for MorApB, Aq.
Zero element The zero element of MorApB, Aqis naturally the zero morphism 0 :BÑA. From the identity σApIdA,0q “IdA proved in lemma 2.3.2 andpa,0q “ pIdA,0qawe geta´0“a.
Inverse From the identityσA∆A“0, proved in lemma 2.3.2, andpa, aq “∆Aawe have a´a“σApa, aq “0.
Therefore
p0´aq `a“ p0´aq ´ p0´aq “ p0´0q ´ pa´aq “0.
Commutativity From the identities
p0´bq ´c“ p0´bq ´ pc´0q “ p0´cq ´ pb´0q “ p0´cq ´b and
0´ p0´dq “ pd´dq ´ p0´dq “ pd´0q ´ pd´dq “ pd´0q ´0“d we get
b`c“b´ p0´cq “ p0´ p0´bqq ´ p0´cq “ p0´0q ´ pp0´bq ´cq
“ p0´0q ´ pp0´cq ´bq “ p0´ p0´cqq ´ p0´bq “c´ p0´bq
“c`b.
Associativity Finally, by using the following 5 identities b` p0´cq “b´ p0´ p0´cqq “b´c, b` p0´bq “b´b“0,
0´ pc´dq “ p0´0q ´ pc´dq “ p0´cq ´ p0´dq “ p0´cq `d 0´ pc`dq “0´ pc´ p0´dqq “ p0´cq ` p0´dq “ p0´cq ´d, pa´bq `d“ pa´bq ´ p0´dq “ pa´0q ´ pb´dq “a´ pb´dq,
we get
pa`bq `d“ pa´ p0´bqq `d“a´ pp0´bq ´dq “a´ pp0´bq ´ p0´ p0´dqq
“a´ pp0´0q ´ pb´ p0´dqqq “a´ p0´ pb`dqq “a` pb`dq.
Composition: To show that addition is bilinear with respect to composition of morphisms, let x: X ÑC and y:AÑY be morphisms inA. Now
pa´bqx“σApa, bqx“σApax, bxq “ax´bx and by lemma 2.3.2 we have
ypa´bq “yσApa, bq “σYpyˆyqpa, bq “σYpya, ybq “ya´yb.
This shows that addition is bilinear under composition of morphisms.
By proposition 2.1.3,Ahas the biproduct ofAandB. SinceA, B, C, andf were arbitrary, we have proven thatA is an additive category.
Corollary 2.3.4. Let A be an abelian category and let f :A ÑB and g: AÑC be morphisms. The following sequence
Ai1f´i2gB‘C c W 0
is exact, wherec:B‘CÑW is the coequalizer ofi1f andi2g. In particular, the coequalizerW here is the pushout of f andg by the dual version of the proof of proposition 1.2.7.
Proof. The morphism c is an epimorphism because it is a coequalizer. By proposition 2.2.14 it suffices to show that cis the cokernel ofi1f´i2g. But the coequalizer W ofi1f andi2g is also the cokernel ofi1f´i2g. Indeed, if d : B‘C Ñ W1 is a morphism such that di1f “ di2g, equivalently dpi1f ´i2gq “ 0, then there is a unique morphismesuch thatec“d. Thus the sequence is exact.
Corollary 2.3.5. LetAbe an abelian category. Then every pushout square is a pullback square. Moreover, pushout of a monomorphism is a monomorphism and pullback of an epimorphism is an epimorphism.
Proof. By duality, it suffices to show that pushout of a monomorphism is a monomorphism, because pushout squares correspond to pullback squares and monomorphisms to epimorphisms in the opposite category.
Letf :XÑY be a monomorphism and g:X ÑZ a morphism. Consider the following commutative diagram
X Y
Y ‘Z
Z W
f i2g´i1f g
i1
i2 c
wherec:Y ‘Z ÑW is the cokernel of i2g´i1f. ThenpW, ci1:Y ÑW, ci2:Z ÑWqis the pushout off andg by the proof of 1.2.7.
Let us show that i2g´i1f is a monomorphism. Let x: D Ñ X be a morphism such that pi2g´i1fqx“0.
Now 0“p1pi2g´i1fqx“ ´f x, so x“0 by the fact that f is a monomorphism. This shows thati2g´i1f is a monomorphism. By 2.2.11 we havei2g´i1f “kerc.
To show that ci2 is a monomorphism, lety : D ÑY be a morphism such that ci2y “0. Then there exists a unique morphismφ:D ÑX such thatpi2g´i1fqφ“i2y. Now´f φ“p1pi2g´i1fqφ“p1i2y “0, so φ“0 by the fact thatf is a monomorphism. Fromi1y “ pi2g´i1fqφ“0, we gety “0, becausei2 is a monomorphism.
This shows thatci2 is a monomorphism.
It remains to show that a pushout square is a pullback square. Keep the notation of the diagram and let y : D Ñ Y and z : D Ñ Z be morphisms such that ci1y “ ci2z. Let q : Y Ñ V be the cokernel of f. Now qf “0g“0, so there is a unique morphismr such that rci1 “q andrci2 “0. Fromqy“rci1y“rci2z“0, we get that there is a unique morphismxsuch that f x“y, becausef “kerq. Nowci2z“ci1y “ci1f x“ci2gx, so z “gxby the fact that ci2 is a monomorphism. Uniqueness of the morphismxfollows from the fact that f is a monomorphism. This shows thatpX, f, gqis the pullback ofpW, ci1, ci2q.