Table Of ContentWEAKLY EXACT CATEGORIES AND THE SNAKE LEMMA
AMIRJAFARI
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0
0 Abstract. Wegeneralizethenotionofanexactcategoryandintroduceweakly
2 exact categories. A proof of the snake lemmainthis general setting isgiven.
Someapplications aregiventoillustratehow onecandohomological algebra
n
inaweaklyexactcategory.
a
J
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Contents
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T
1. Introduction 1
C
2. Weakly Exact Category 3
.
h 3. The Snake Lemma 4
t 4. The 3×3 Lemma 6
a
m 5. Examples 8
6. Applications 10
[
7. Additive Weakly Exact Categories 12
1 References 14
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3
2 1. Introduction
.
1 Short and long exact sequences are among the most fundamental concepts in
0
mathematics. The definition of an exact sequence is based on kernel and cokernel,
9
which can be defined for any category with a zero object 0, i.e. an object that is
0
: both initial and final. In such a category, kernel and cokernel are equalizer and
v
coequalizer of:
i
X f
r A ////B .
a 0
Kerneland cokerneldo not need to exist in general,but by definition if they exist,
they are unique up to a unique isomorphism. As is the case for any equalizer and
coequalizer, kernel is monomorphism and cokernel is epimorphism. In an abelian
categorythe converse is also true: any monomorphismis a kerneland any epimor-
phism is a cokernel. In fact any morphism A → B factors as A → I → B with
A → I a cokernel and I → B a kernel. This factorization property for a cate-
gory with a zero object is so important that together with the existence of finite
products, imply that the category is abelian (see 1.597 of [FS]).
A short exact sequence in an abelian category is a sequence of morphisms:
A−→i B −p→C
such that i is a kernel of p and p is a cokernel of i. This definition makes sense for
any category with a zero object and we call such sequences special. However if we
1
2 AMIRJAFARI
let all special sequences to be short exact, the nice properties of exact sequences
fail to extend unless we either restrict this definition or restrict the categories by
putting extra conditions.
In the first approach we take a class of special sequences that satisfy certain
carefullychosenaxiomsandcallthemshortexactsequences. Suchanapproachfor
an additive category was considered by Quillen in [Q], and it has its predecessors
e.g. in Heller [H]. In the second approach we assume that the category satisfies
certain carefully chosen axioms (that are modeled on the categories of groups,
rings, Lie algebras,etc.) and we take all special sequences to be short exact. Such
an approach leads to the notions like Barr-exact, sequential, homological or semi-
abelian categories , see [BB] and the references therein.
Once we have the notion of a short exact sequence, one can easily define a long
exact sequence.
Definition 1.1. A long sequence
...−→Ai−1 −→Ai −→Ai+1 −→...
is exact if every morphism Ai−1 → Ai factors as Ai−1 → Zi → Ai such that
Z → A → Z are short exact for all i. If the sequence is finite from one or
i i i+1
both ends we also assume that the first A → Z sits in a short exact sequence
i i+1
Z → A → Z , and similarly the last Z → A sits in a short exact sequence
i i i+1 i i
Z →A →Z .
i i i+1
Among the consequences of exactness, an important result that is of constant
use in homological algebra is the snake lemma:
Lemma1.2. Inanabeliancategory,forthecommutativediagram below,withexact
rows:
A φ1 // A φ2 // A //0
1 2 3
f1 f2 f3
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // B //B //B
1 φ′1 2 φ′2 3
there is a natural1 connecting morphism δ : ker(f ) → coker(f ) that makes the
3 1
sequence
ker(f )→ker(f )→ker(f )→δ coker(f )→coker(f )→coker(f )
1 2 3 1 2 3
exact. The maps between the kernels and the cokernels arise by their universal
property. Furthermore if the top row is short exact then 0→ker(f )→ker(f ) is
1 2
exact and if the bottom row is short exact then coker(f )→coker(f )→0 is exact.
2 3
1Natural here has a precise meaning: Given another diagram as above with A′i, Bi′ and fi′,
and morphisms between them, making all squares commutative, we should have a commutative
diagram:
ker(f3) δ //coker(f1)
ker((cid:15)(cid:15)f3′) δ′ //coker(cid:15)(cid:15)(f1′)
WEAKLY EXACT CATEGORIES AND THE SNAKE LEMMA 3
The common proofs of this lemma use a method called chasing the elements.
This can be applied for concrete categories, like abelian groups or more generally
modulesoveraring. Thegeneralcasecanbededucedusingtheembeddingtheorem
thatassertsanysmallabeliancategorycanbeembeddedinthecategoryofmodules
over a ring. There are several natural proofs without the use of elements in the
literature, see e.g. [FHH].
Inthisnote,weconsidercertainclassesofspecialsequencesinageneralcategory
with a zero object, that allow us to prove this lemma in a more general setting.
2. Weakly Exact Category
In this section we define a weakly exact category. These are categories with a
zero object and certain class of morphisms, called deflation. deflations have kernel
and are cokernel of their kernel. Hence if p : B → C is a deflation and i : A → B
is its kernel then:
A−→i B −p→C
is a special sequence. We call such sequences short exact. A kernel of a defla-
tion is called an inflation. Note that a deflation is epimorphism and an inflation
is monomorphism. The terminology of admissible epimorphism and admissible
monomorphism is also used for deflation and inflation. As in the introduction, we
can also define long exact sequences in a weakly exact category.
The following axioms are very similar to Heller’s axioms for an exact additive
category, see appendix B of [B] or [H]. The difference is that we do not assume
inflations are closed under composition (and hence the definition is not self dual),
alsoaxiom3isweakerthanitscounterpartinHeller’saxioms. Ofcoursethemajor
difference is that we no longer assume that our category is additive. The author
doesnotknowofanynon-trivialexampleofaselfdualweaklyexactcategorythatis
notadditive. Howeverthe following definition makes the categoryof groups,rings,
Liealgebras,etcintoaweaklyexactcategory,ifwetakeall(regular)epimorphisms
to be deflation.
Definition 2.1. A weakly exact category is a category with a zero object and a
collection of morphisms called deflations satisfying the following axioms.
(0) Any isomorphism and any morphism A→0 is a deflation.
(1) Any deflation has a kernel and is cokernel of its kernel. The kernel of a
deflation is called an inflation.
(2) The class of deflations is closed under composition.
(3) For two composable morphism f and g if gf and f are deflation then g is
a deflation.
(4) (3 by 3 lemma) In the following commutative diagram with short exact
columns if the second and the third rowsare shortexact then the first row
is short exact.
A //A //A
1 2 3
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
B //B //B
1 2 3
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
C // C //C
1 2 3
4 AMIRJAFARI
Lemma 2.2. Axiom 4 is self dual, i.e. in the diagram above with short exact rows,
if the first and second columns are short exact then the third column is short exact.
Proof. ThemorphismB →C isadeflationbyaxiom2and3. SoletA′ →B
3 3 3 3
beitskernel. BypropertiesofkernelwehaveamapfromA →A′ whichmakesthe
2 3
diagramabovewithA replacedwithA′ commutative. Itscolumnsandthesecond
3 3
and the third rows are short exact, hence axiom 3 implies that A → A → A′ is
1 2 3
shortexact. SincebothA andA′ arecokernelofA →A ,thereisanisomorphism
3 3 1 2
A → A′ such that the composite A → A′ → B is a kernel for B → C , hence
3 3 3 3 3 3 3
A →B →C is short exact. (cid:3)
3 3 3
3. The Snake Lemma
In this section we will prove:
Theorem3.1. (Thesnakelemma)Inaweaklyexactcategory,givenacommutative
diagram
(1) A φ1 // A φ2 // A
1 2 3
f1 f2 f3
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
B //B //B
1 φ′1 2 φ′2 3
where the rows are short exact and f ’s are admissible, i.e they have a factorization
i
A −f→i′ I −f→i′′ B
i i i
with f′ a deflation with kernel k : K → A and f′′ an inflation with cokernel
i i i i i
c :B →C , there is a natural connecting morphism δ :K →C such that
i i i 3 1
K −ψ→1 K −ψ→2 K −δ→C −ψ→1′ C −ψ→2′ C
1 2 3 1 2 3
is exact and ψ is an inflation and ψ′ is adeflation. Hereψ and ψ′ arethe natural
1 2 i i
morphisms induced by φ and φ′.
i i
Proof. This proof is inspired by the proof of the snake lemma for a sequential
category given in [Bo]. First observe that by axioms 2 and 3, ψ′ is a deflation.
2
By the universal properties of kernel and cokernel there is a unique morphism
φ′′ :I →I , such that the following diagram is commutative:
2 2 3
A φ2 // A
2 3
f2′ f3′
I(cid:15)(cid:15) φ′2′ //I(cid:15)(cid:15)
2 3
f2′′ f3′′
B(cid:15)(cid:15) φ′2 // B(cid:15)(cid:15)
2 3
WEAKLY EXACT CATEGORIES AND THE SNAKE LEMMA 5
By axioms 2 and3, φ′′ is a deflation and let φ′′ :J →I be its kernel. Then by
2 1 1 2
axiom4, the sequence J →B →ker(ψ′) in the following diagramis short exact.
1 1 2
(2) J ǫ //B c′2 // ker(ψ′)
1 1 2
φ′1′ φ′1 i
I(cid:15)(cid:15) f2′′ //B(cid:15)(cid:15) c2 //C(cid:15)(cid:15)
2 2 2
φ′2′ φ′2 ψ2′
I(cid:15)(cid:15) f3′′ //B(cid:15)(cid:15) c3 //C(cid:15)(cid:15)
3 3 3
Note that since the composite C −ψ→1′ C −ψ→2′ C is trivial ψ′ factors through
1 2 3 1
ker(ψ′) via a morphism α:C →ker(ψ′). Since αc =c′ and both c and c′ are
2 1 2 2 2 2 2
deflations,itfollowsthatαis a deflation. This showsthatC →C →C is exact.
1 2 3
Consider the following diagram
(3) I1 ǫ′ // T j //ker(ψ1′)
ǫ i′
I f1′′ // B(cid:15)(cid:15) c1 //C(cid:15)(cid:15)
1 1 1
ψ1′
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // ker(ψ′) ker(ψ′)
2 2
whereT isthe kernelofψ′c =c φ′. Axiom4againimplies thatthe firstrowis
1 1 2 1
a short exact sequence. Since in diagram (2), c φ′ = ic′, we can identify T →B
2 1 2 1
with J → B . Let π : P → A be the kernel of f′φ : A → I or equivalently
1 1 1 2 3 2 2 3
the kernel of φ′′f′ :A →I . It fits in the following commutative diagrams:
2 2 2 3
A1 θ1 //P θ2 // K3 K2 k2′ //P p // J1
π1 i π1 φ′1′
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
A // A //A K //A //I
1 φ1 2 φ2 3 1 k2 2 f2′ 2
f3′ φ′2′
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //I I 0 //I I
3 3 3 3
By axiom 4 the first rows are short exact. Consider the commutative diagram:
A1 θ1 // P θ2 // K3
f1′ p δ′
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
I //J // ker(ψ′)
1 ǫ′ 1 j 1
Toshowthatpθ =ǫ′f′ itisenoughtoshowǫpθ =ǫǫ′f′. Nowaneasychaseinthe
1 1 1 1
diagrams show that both sides are f . We define δ : K −δ→′ ker(ψ′) ֒→i′ C . Note
1 3 1 1
thatsincep,j andθ aredeflations,axiom2and3implythatδ′ isadeflation. This
2
6 AMIRJAFARI
shows that K → C → C is exact. Finally if we apply axiom 4 to the diagram
3 1 2
above, we deduce that
K →K →ker(δ′)
1 2
is a short exact sequence. The theorem is proved. (cid:3)
Lemma 3.2. Let f : A → B and g : B → C be two morphisms. If gf and g are
inflation then f is an inflation.
Proof. Let B −g→ C −p→ D and A −g→f C −q→ D′ be short exact sequences.
Consider the following diagram:
gf q
A //C // D′
p p′
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // D D
The left square commutes since fg factors through the kernel of p, i.e. B. The
morphism p′ exists by universal property of cokernel. Now axiom 3 implies that p′
is a deflation. So we can apply axiom 4 and deduce that A →f B → ker(p′) is a
short exact sequence so f is an inflation. (cid:3)
Remark 3.3. IfinthestatementofthesnakelemmawecomposeA →A witha
1 2
deflationA′ →A andcomposeB →B withaninflationB →B′,wegetnatural
1 1 2 3 3 3
morphismsK′ →K andC →C′ whereK′ isthekernelofA′ →A →B andC′
1 1 3 3 1 1 1 1 3
is the cokernelof A →B →B′. It follows easily from the axioms that K′ →K
3 3 3 1 1
is a deflation and C → C′ is an inflation (for this we need to use the previous
3 3
lemma). Hence we have a long exact sequence K′ →K →K →C →C →C′.
1 2 3 1 2 3
4. The 3×3 Lemma
In this section we show that axiom 4 follows from the following two axioms:
(4a) The pullback of a deflation along a deflation exists and is a deflation.
(4b) Given a commutative diagram:
A φ1 // A
1f1 2f2BBBBBφBB2B
(cid:15)(cid:15) (cid:15)(cid:15)
B //B // A
1 φ′1 2 φ′2 3
where φ and φ′ are deflation and φ and φ′ are their kernels, f is a deflation if
2 2 1 1 1
and only if f is a deflation.
2
These are usually easier to check in a category. However axiom 4 does not in
general imply (4a) and (4b).
Axiom 4, says in the commutative diagram:
(4) A φ1 // A φ2 // A
1 2 3
f1 f2 f3
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
B //B //B
1 φ′1 2 φ′2 3
WEAKLY EXACT CATEGORIES AND THE SNAKE LEMMA 7
where the rows are short exact and f ’s are deflations with kernel K , the induced
i i
sequence K →K →K is short exact.
1 2 3
Consider the pullback of φ′ along f :
2 3
>> P φ′2′ //A3
φ′1′
π2 f3
(cid:15)(cid:15) (cid:15)(cid:15)
B //B //B
1 φ′1 2 φ′2 3
Since φ′f = f φ , by the universal property of pullback, there is a unique mor-
2 2 3 2
phism π :A →P such that π π =f and φ′′π =φ . Also since the kernel of a
1 2 2 1 2 2 1 2
morphism and its pullback are isomorphic, we have a lift of the kernel φ′ of φ′ to
1 2
φ′′ : B → P which is a kernel of φ′′. We therefore get the following commutative
1 1 2
diagram:
A φ1 //A
1f1 2π1BBBBBφBB2B
B(cid:15)(cid:15)1BφBB′1φBB′1′BBB //BP(cid:15)(cid:15)(cid:15)(cid:15)π2 φφ′2′2′ //// BA(cid:15)(cid:15)3f3
2 3
Now by axiom 4b, π is a deflation whose kernel is k′ :K →k1 A →φ1 A . Similarly
1 1 1 1 2
the kernel of f , k : K → A has a unique lift k′ : K → P as a kernel of π .
3 3 3 3 3 3 2
Note that π is a deflation by axiom 4a. Finally by axiom (4b) the commutative
2
diagram:
K k2 //A
2ψ2 2π1BBBBBfB2BB
(cid:15)(cid:15) (cid:15)(cid:15)
K3 k3′ // P π2 //B2
implies that ψ is a deflation whose kernel is the same as π , i.e. K →ψ1 K hence
2 1 1 2
thesequenceK →K →K isashortexactsequence. Forcompletenessweprove
1 2 3
the full version of the 3 by 3 lemma:
Lemma 4.1. In a weakly exact category satisfying (4a) and (4b), given a commu-
tative diagram:
A φ1 // A φ2 // A
1 2 3
f1 f2 f3
B(cid:15)(cid:15) φ′1 // B(cid:15)(cid:15) φ′2 //B(cid:15)(cid:15)
1 2 3
g1 g2 g3
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
C // C //C
1 φ′1′ 2 φ′2′ 3
where all the columns and the first and the last rows are short exact and φ′φ′ =0,
2 1
then the middle row is short exact.
8 AMIRJAFARI
Proof. Consider the following pullback:
P θ2 //C2
ψ2 φ′2′
(cid:15)(cid:15) (cid:15)(cid:15)
B //C
3 g3 3
As in the previous proof we get morphisms θ : A → P as kernel of θ and
1 3 2
ψ :B →P suchthatψ ψ =φ′ suchthatthe followingdiagramis commutative:
1 2 2 1 2
A f2 //B
φ2 2 ψ1 2BBBBBgB2BB
A(cid:15)(cid:15)3BfBB3θBB1BBψB //2P(cid:15)(cid:15)(cid:15)(cid:15) θ2 // C(cid:15)(cid:15)2φ′2′
B // C
3 g3 3
Since φ′′ and φ are deflations, by axiom (4a) and(4b) ψ and ψ are deflations
2 2 2 1
and hence φ′ = ψ ψ is a deflation. Let φ˜′ : B′ → B be its kernel. By axiom
2 2 1 1 1 2
(4) the induced sequence A −f→1′ B′ −g→1′ C is short exact. Since φ′φ′ = 0 the
1 1 1 2 1
morphismφ′ factorsthroughφ˜′ viaψ :B →B′. Wehaveacommutativediagram:
1 1 1 1
B
A }f}}1f}}1′}}}//>>B(cid:15)(cid:15)1′ψAAAgAA1′gAA1A// C
1 1 1
Now by axiom (4b), ψ is a deflation and since
A f1 //B
1 1
ψ
(cid:15)(cid:15)
A //B′
1 f1′ 1
isapullback,kernelofψisthesameasthekerneloftheidentityofA ,andhenceis
1
zero. So 0→B →ψ B′ is short exact i.e. ψ is an isomorphism. Therefor B →B
1 2 1 2
is a kernel of B →B and the middle row is short exact. (cid:3)
2 3
5. Examples
Recall that a morphism that is coequalizer of two parallel arrows is called a
regular epimorphism. For example any cokernel is a regular epimorphism. In a
weakly exact category, any deflation being a cokernel is regular epimorphism. We
make the following definition:
Definition 5.1. A categorywith a zeroobject is saidto be weaklyhomological,if
theclassofallregularepimorphismssatisfytheaxiomsforaweaklyexactcategory.
Obviously this is a the maximal class.
WEAKLY EXACT CATEGORIES AND THE SNAKE LEMMA 9
For example any abelian category is weakly homological. In this case every
epimorphism is regular.
Recall that a regular category is a category such that
(1) All finite limits exists.
(2) The class of regular epimorphisms is closed under arbitrary pullbacks.
(3) For any morphism f :X →Y the kernel pair:
X×Y X p1 ////X .
p0
has a coequalizer g : X →Z and therefore f = hg for a unique morphism
h:Z →Y.
It is shown that h is a monomorphism. So any morphism can be factored as a
regular epimorphism followed by a monomorphism, in a unique way.
Lemma 5.2. A regular category with zero object is weakly homological if and only
if every regular morphism is cokernel of its kernel.
Proof. The ”only if” part is clear, since axiom 1 should hold for all regular
epimorphisms. To prove the other direction, first note that in a regular category
axioms0,2and3hold(seeappendixA.5of[BB]). Actuallyforaxiom3,itisenough
to know that gf is a regular epimorphism. Also axiom 4a and half of 4b (if f is
2
regular epic then f is) are true, since regular epics are stable under pullbacks. It
1
remains to show the other half of 4b. Factor f as A −f→2′ P −f→2′′ B , with f′
2 2 2 2
regularepicandf′′ monic. sincef andf′ arecokerneloftheir isomorphickernels,
2 1 2
by universal properties of cokernel we get a morphism φ′′ : B → P making the
1 1
following diagram commutative:
A φ1 // A
1f1 2f2′BBBBBφBB2B
(cid:15)(cid:15) (cid:15)(cid:15)
B1 φ′1′ //P φ′2′ // A3
f2′′
(cid:15)(cid:15)
B //B // A
1 φ′1 2 φ′2 3
Since f′′ is monic, φ′′ is kernel of φ′′ and therefore the lower right square is a
2 1 2
pullback. Hence f′′ is a deflation. And therefore f =f′′f′ is a deflation. (cid:3)
2 2 2 2
We give an example of a category, where every regular epimorphism is cokernel
of its kernel.
Lemma5.3. Inanyadditivecategorywithfinitelimits, anyregularepiciscokernel
of its kernel.
Proof. Let f : X → Y be a regular epimorphism with kernel k : K → X and
kernel pair X ×Y X p1 ////X . Since fp0 = fp1 hence f(p0 −p1) = 0 and hence
p0
p −p factors through k. Therefore for g : X → Z if gk = 0 then g(p −p ) = 0
0 1 0 1
andhence g coequalizes p an p . Since f is the universalcoequalizerof p andp ,
0 1 0 1
g factors uniquely through f, this is the definition of a cokernel for k. (cid:3)
10 AMIRJAFARI
More generally, it is shown in [Bo], that in any protomodular category with a
zeroobject,anyregularepiciscokernelofitskernelandhencearegularprotomod-
ular category with a zero object is weakly homological. Such a category is called
homologicalin [BB]. A category is Protomodular if for any pullback diagram:
φ′
A′ //B′
f g
(cid:15)(cid:15) (cid:15)(cid:15)
A // B
φ
where f and g are split epic, i.e. we have s : A → A′ and t : B → B′ such that
fs= id and gt= id , the pair (φ′,t) is jointly strongly epic. Only the following
A B
special case of this definition is needed to show any regular epic is cokernel of its
kernel. Given a regular epic f : X → Y with kernel k : K → X and kernel pair
X×Y X p1 ////X , we get the following pullback diagram:
p0
K×K k′ //X × X
Y
q0 q1 p0 p1
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
K k //X
of split epic morphisms, where s:K →K×K and t:X →X × X are given by
Y
the diagonals. Then the pair (k′,t) is jointly epic.
6. Applications
In this section we discuss several applications of the snake lemma, similar to
those in homological algebra on abelian categories. But instead we work in a
weakly exact category. We will consider chain complexes in such a category, i.e.
sequences ··· → Ai−1 → Ai → Ai+1 → ... where the composition of successive
arrows are zero. We have a natural definition for morphisms between such chain
complexes. The morphisms A → A are called differentials and if they are
i i+1
admissible morphisms then the chain complex is called an admissible complex.
Definition 6.1. The cohomology of a chain complex A →f B →g C with inflation
f and deflation g is defined by one the following two ways:
(1) Let A →f B →g′ C′ be a short exact sequence, by properties of cokernel g
factors through C′: B →g′ C′ →C. By axiom 3, C′ →C is a deflation and
its kernel H is the cohomology.
(2) Let A′ →f′ B →g C be a short exact sequence, by properties of kernel f
factors through A′ : A→A′ →f′ B. By lemma 3.2., A→A′ is an inflation
and its cokernel H′ is the cohomology.
Lemma 6.2. With the notation of the previous definition, there is a natural mor-
phism H → H′ that is both deflation and inflation and hence an isomorphism.
Furthermore the complex A→B →C is exact if and only if H =0.
