THE SNAIL LEMMA ENRICO M. VITALE

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1 THE SNIL LEMM ENRICO M. VITLE STRCT. The classical snake lemma produces a six terms exact sequence starting rom a commutative square with one o the edge being a regular epimorphism. We establish a new diagram lemma, that we call snail lemma, removing such a condition. We also show that the snail lemma subsumes the snake lemma and we give an interpretation o the snail lemma in terms o strong homotopy kernels. Our results hold in any pointed regular protomodular category. Contents 1 Introduction 1 2 The snake lemma 2 3 The snail lemma 4 4 Comparing the snake and the snail categorical explication o the snail lemma 12 6 The complete snail lemma Introduction One o the basic diagram lemmas in homological algebra is the snake lemma (also called kernel-cokernel lemma). In an abelian category, it can be stated in the ollowing way : rom the commutative diagram Ker() k (1) K() Ker( ) k and under the assumption that is an epimorphism, it is possible to get an exact sequence Ker(K()) Ker() Ker() Cok(K()) Cok() Cok() 2 Mathematics Subject Classiication: 18G5, 18D5, 18G55. Key words and phrases: snail lemma, snake lemma, protomodular category, strong homotopy kernel. 1

2 I we replace epimorphism with regular epimorphism and i, and K() are proper morphisms (Deinition 2.2) the snake lemma holds also in several important non abelian categories, like groups, crossed modules, Lie algebras, not necessarily unitary rings. Despite its very clear ormulation, the snake lemma is somehow asymmetric because o the hypothesis on the morphism. The aim o this paper is to study what happens i we remove such condition. What we prove is that we can get a six terms exact sequence (the snail sequence) starting rom any commutative diagram like 2 (2) (with and proper i we work in a non abelian context); moreover, the snail sequence coincides with the usual snake sequence whenever is a regular epimorphism. In order to get the snail sequence, we have to replace the kernels appearing in diagram (1) with a dierent (very simple) construction, which in act is a kind o 2-dimensional kernel o diagram (2). More in detail, the layout o this paper is as ollows : in Section 2 we recall the snake lemma in its general orm established by D. ourn in the context o pointed regular protomodular categories (= homological categories, in the terminology o [3]), as well as some other results rom [2]. general reerence or protomodular categories is [3], a more concise introduction can be ound in [4]; abelian categories as well as groups and all the other examples quoted above are pointed regular protomodular categories. Pointed regular protomodular categories are also the ramework o Section 3, which is completely devoted to state and prove the snail lemma, and o Section 4, where we show that the snail lemma subsumes the snake lemma. It is worthwile to note that, in order to compare the snail sequence and the snake sequence, we need an intermediate result (Lemma 4.1) which is very close to the characterization o subtractive categories established in [8]. The precise relation between the snail lemma and the axiomatic or subtractive categories will be explaineded in [9]. In Section 5 we give a precise explication o the 2-categorical meaning o the construction involved in the snail lemma (in contrast with the construction involved in the snake lemma, which is a categorical kernel but does not satisy any 2-dimensional universal property). Finally, in Section 6 we specialize the situation to abelian categories : in this case, by duality we easily get a longer version o the snail sequence. We use diagrammatic notation or composition : g C is written g. 2. The snake lemma Let be a pointed, regular and protomodular category.

3 Notation 2.1 Here is the notation we use or kernels, cokernels, and the induced morphisms. Ker() K() Ker( ) k c Cok() C() k c Cok( ) To start, we recall the snake lemma as proved by D. ourn in [2]. Deinition 2.2 morphism : in is proper i it admits a cokernel c and the actorization o along the kernel o c is a regular epimorphism c Cok() k c Ker(c ) Snake Lemma 2.3 Consider the ollowing commutative diagram in and assume that and are regular epimorphisms 3 Ker() k K() Ker( ) k I, and K() are proper, then there exists a morphism d: Ker() Cok() such that the ollowing sequence is exact Ker(K()) K(k ) Ker() K() Ker() d Cok(K()) C(k ) Cok() C( ) Cok() Observe that i the category is abelian, then any morphism in is proper. This is the reason why there are no assumptions on, and K() or the snake lemma in an abelian category. On the way to prove the snake lemma, ourn establishes in [2] the ollowing acts that we need later. Proposition 2.4 Consider the ollowing commutative diagram in and assume that is a regular epimorphism Ker() K() Ker() Ker() k k k K() Ker( ) k

4 4 1. I K() is a regular epimorphism, then K() is a regular epimrphism. 2. I K() and are regular epimorphisms, then is a regular epimrphism. Proposition 2.5 morphism : in is a monomorphism i and only i its kernel k : Ker() is the zero morphism. 3. The snail lemma In this section, is a pointed, regular and protomodular category. Consider a commutative diagram in and construct the ollowing diagram, where : -, is the pullback o and, - =, is the unique morphism such that = and =, - k =, k is the unique morphism such that k = and k = k, and it is a kernel o because is a pullback o, - C( ) is the unique morphism such that c C( ) = c. Ker( ) = Ker() k k, c Cok() c Cok() C( )

5 5 Snail Lemma 3.1 Consider the ollowing commutative diagram in Ker() k, c Cok() K(id) id C( ) Ker() K() Ker() k k c c Cok() Cok() C( ) I, and are proper, then the ollowing sequence is exact Ker() K(id) Ker() K() Ker() k c Cok() C( ) Cok() C( ) Cok() Proo. Exactness in Ker(): obvious because K(id): Ker() Ker() is a kernel o K() by interchange o limits. Exactness in Ker(): irst, observe that K() k = k (compose with the pullback projections) and, thereore, K() k c =. Now, to prove the exactness in Ker() we use the ollowing diagram, where π is the unique morphism such that π k (k c ) = K(). Observe that K(k ) is a monomorphism because K(k ) k c = k (k c ) k and k (k c ) and k are monomorphisms. Ker(k c ) k (k c) π K() Ker() Ker() k c K(k ) k Ker(c k ) Cok() k k c id, c Cok()

6 We have to prove that π is a regular epimorphism. This ollows rom the act that is a regular epimorphism (because is proper) and the ollowing square is a pullback. 6 Ker() k π Ker(k c ) K(k ) Ker(c ) To check its commutativity, compose with the monomorphism k c. To check its universality, consider morphisms x: X and y: X Ker(k c ) such that x = y K(k ). We have : x = x = x k c = y K(k ) k c = y k (k c ) k = y k (k c ) = Thereore, there exists a unique z: X Ker() such that z k = x. The condition z π = y ollows rom z k = x because K(k ) is a monomorphism. Exactness in Cok(): irst, observe that k c C( ) = k c =. Now, to prove the exactness in Cok() we use the ollowing commutative diagram. Ker(c ) Ker( K( ) k ) = Ker() K(c ) k k c Ker(c ) Ker(C( )) k C( ) k c, c Cok() c Cok() C( ) We have to prove that K(c ) is a regular epimorphism. For this, observe that K( ) = (indeed, composing with the monomorphism k c, both give ). Since is proper, is a regular epimorphism, and thereore the equation K( ) = implies that K( ) is a

7 regular epimorphism. We can conclude that K(c ) is a regular epimorphism by applying point 1 o Proposition 2.4 to the ollowing diagram. 7 Ker( ) K(c ) Ker(C( )) Ker(c ) k c k, c k C( ) Cok() K( ) Ker(c ) C( ) k c c Cok() Exactness in Cok(): irst, observe that c C( ) C( ) = and then, since c is an epimorphism, we have C( ) C( ) =. Now, to prove the exactness in Cok() we use the ollowing commutative diagram, where π is the unique morphism such that π k C( ) = C( )., c Cok() π C( ) c Ker(C( )) c k C( ) Cok() Cok() C( ) k c Ker(c ) We have to prove that π is a regular epimorphism. For this, we split the construction o the pullback, in two steps :, x z,k c Ker(c ) Ker(c ) y k c

8 Since is proper, is a regular epimorphism, and then x also is a regular epimorphism because it is a pullback o. Moreover, since y c C( ) =, there exists a unique t:,k c Ker(c ) Ker(C( )) such that y c = t k C( ). Now observe that the ollowing square commutes 8, x,k c Ker(c ) c t Cok() π Ker(C( )) (or this, compose with the monomorphism k C( )). Thereore, since x is a regular epimorphism, to prove that π is a regular epimorphism it remains to show that t is a regular epimorphism. This can be done using the ollowing commutative diagrams z,k c Ker(c ) Ker(c ) y (1) k c c Cok() (2),k c Ker(c ) t Ker(C( )) 1 y (3) C( ) c Cok() (4) C( ) Cok() Since (1) and (2) are pullbacks, so is (1)+(2), that is, y is a kernel o c = c C( ). Thereore, (3)+(4) is a pullback, but also (4) is a pullback, and then (3) is a pullback. Since c is a regular epimorphism, this proves that t is a regular epimorphism. 4. Comparing the snake and the snail s in the previous sections, is a pointed, regular and protomodular category. We need a preliminary result. Lemma 4.1 Consider the ollowing diagram in Ker(x) Ker(y) k y k x P y Y I x and k x y are regular epimorphisms, then k y x is a regular epimorphism. x X

9 Proo. Consider the unique actorization x : y : P X Y o x and y through the product. y applying point 2 o Proposition 2.4 to the diagram 9 Ker(x) k x P x X k x y Y :1 x:y X Y πx X id we get that x : y is a regular epimorphism. Consider now the ollowing commutative diagram Ker(y) k y P x:y k y x (1) 1: X (2) X Y π Y Y Since (2) and (1)+(2) are pullbacks, (1) also is a pullback. Thereore, k y x is a regular epimorphism because it is a pullback o x : y. We will also use the ollowing simple act, which holds i is pointed and has kernels. Lemma 4.2 Consider the ollowing commutative diagram in Ker() k K() Ker( ) k I is a monomorphism, then the let-hand square is a pullback. I we compare the snake sequence (Lemma 2.3) and the snail sequence (Lemma 3.1) Ker(K()) K(k ) Ker() K() Ker() d Cok(K()) C(k ) Cok() C( ) Cok() Ker() K(id) Ker() K() Ker() k c Cok() C( ) Cok() C( ) Cok() we see that the only dierence is that Cok(K()) is replaced, in the snail sequence, by Cok(). Indeed, by exchange o limits, both K(k ): Ker(K()) Ker() and K(id): Ker() Ker() are kernels o K(): Ker() Ker(). In order to show that Cok(K()) and Cok() are isomorphic, let us put together the construction underlying the snake lemma and the construction underlying the snail lemma

10 in the ollowing diagram, where : - k = k, is the unique morphism such that k = k and k =, and it is a kernel o because is a pullback o, - C(k ) is the unique morphism such that k c = c K() C(k ) (such a morphism exists because k = K() k, as one easily cheks by composing with the pullback projections). 1 k Ker(), c K() k Cok() Ker( ) = Ker( ) k C(k ) c K() Cok(K()) Proposition 4.3 (With the previous notation.) 1. I and K() are proper, then C(k ) is a monomorphism; 2. I is a regular epimorphism, then C(k ) is a regular epimorphism. Proo. Proo o 1. y Lemma 4.2, the let-hand square in the ollowing commutative diagram is a pullback Ker() k K() Ker( ) k, id Thereore, pulling back k along the (regular epi, mono)-actorization o gives the (regular epi, mono)-actorization o K(). Since and K() are proper, this means that

11 11 in the ollowing diagram squares (1) and (2) are pullbacks. Ker() K() k (1) K(k ) (2) Ker(c K() ) Ker(c ) k ck() k c Ker( ), k (3) c K() c Ker(C(k )) Cok(K()) Cok() k C(k ) C(k ) Now we prove that (3) also is a pullback, so that is a regular epimorphism and then k C(k ) is zero, which implies that C(k ) is a monomorphism (Proposition 2.5). Clearly, (3) commutes. Consider morphisms x: X Ker(C(k )) and y: X Ker( ) such that x k C(k ) = y c K(). We have y k c = y c K() C(k ) = x k C(k ) C(k ) = x = so that there exists z: X Ker(c ) such that z k c = y k. Since (2) is a pullback, this implies that there exists t: X Ker(c K() ) such that t K(k ) = z and t k ck() = y. Moreover, t k C(k ) = t k ck() c K() = y c K() = x k C(k ), and then t = x because k C(k ) is a monomorphism. Proo o 2. Consider the ollowing commutative diagram Ker(c ) Ker(k ) k k c, c K() Cok(K()) C(k ) c Cok() Since is a regular epimorphism and = = k c, we have that k c is a

12 12 regular epimorphism. Thereore, we can apply Lemma 4.1 to Ker(c ) Ker(k ) k k c, c Cok() and we have that k c is a regular epimorphism. Since k c = c K() C(k ), we can conclude that C(k ) is a regular epimorphism categorical explication o the snail lemma The most economical way to explain the construction involved in the snail lemma is by using strong homotopy kernels. Indeed, to express the universal property o a strong homotopy kernel we only need null-homotopies. In order to ormalize null-homotopies and (strong) homotopy kernels, we adopt the ollowing setting, introduced in [7]. Deinition 5.1 category with null-homotopies is given by - a category, - or each morphism : in, a set N() (the set o null-homotopies on ), - or each triple o composable morphisms :, g: C, h: C D, a map h: N(g) N( g h), µ µ h (I = id or h = id C, we write µ h or µ instead o µ h.) These data have to satisy the ollowing associativity condition : given morphisms g C h D h D then or any µ N(g) one has ( ) µ (h h ) = ( µ h) h. Deinition 5.2 Let be a category with null-homotopies and let : be a morphism in. triple Ker(), K(): Ker(), k() N(K() ) 1. is a homotopy kernel o i or any triple D, g: D, µ N(g ), there exists a unique morphism g : D Ker() such that g K() = g and g k() = µ 2. is a strong homotopy kernel o i it is a homotopy kernel o and, moreover, or any triple D, h: D Ker(), µ N(h K()) such that µ = h k(), there exists a unique λ N(h) such that λ K() = µ.

13 Example 5.3 To help intuition, here is an easy example. The category is the category Grpd o groupoids with a distinguished object and pointed unctors (that is, unctors preserving the object ). For a pointed unctor :, the set N() is the set o natural transormations λ: such that λ = id, where : is the constant pointed unctor. The operation µ g is the reduced horizontal composition o natural transormations. The strong homotopy kernel o : is the usual comma (or slice) groupoid / whose objects are pairs (a, b 1 : (a ) ). In the rest o this section, is a category with pullbacks. Notation 5.4 Let us describe the category with null-homotopies rr(). - n object is a morphism : in. - morphism F : is a pair (, ) o morphisms in such that the ollowing diagram commutes - Given a morphism F : in rr(), the set o null-homotopies N(F) is the set o morphisms λ: in such that λ = and λ =. - Given three morphisms F :, G: C, H : C D in rr(), the operation F H : N(G) N(F G H) is given by 13 g C µ g C g C h D µ h δ g C D h The irst part o the next proposition is obvious (it requires that is pointed). It is included in the statement just to stress the relation between snails, snakes, and kernels. Proposition 5.5 Let F = (, ): be a morphism in rr(). 1. The construction involved in the snake sequence Ker() k K() Ker( ) k is the kernel o F in the category rr().

14 14 2. The construction involved in the snail sequence, id is the strong homotopy kernel o F in the category with null-homotopies rr(). Proo. Proo o 2. (Homotopy kernel) Consider a triple D, G: D, µ N(G F) in rr() D g δ µ D g Since µ = g, there exists a unique morphism µ: D, such that µ = µ and µ = g. We get in this way the ollowing morphism G : D K(F) D δ D µ g, Conditions G K(F) = G and G k(f) = µ are satisied : the irst one amounts to g id = g and µ = g, and the second one amounts to µ = µ. s ar as the uniqueness o the actorization G is concerned, given a morphism H = (h, h ): D Ker(F), condition H K(F) = G means that h = g and h = g, and condition H k(f) = µ means that h = µ. Thereore, h = µ and then H = G. (Strong homotopy kernel) Consider a triple D, H : D Ker(F), µ N(H K(F)) in rr() h id D δ µ D h, satisying the condition µ F = H k(f), which amounts to µ = h. For a null-homotopy λ N(H), the condition λ K(F) = µ means λ id = µ. This gives the uniqueness o the null-homotopy λ, and it remains just to check that h D µ δ D h,

15 is a null-homotopy. In particular, to check that µ = h, compose with the pullback projections (and use µ = h when composing with ). Remark 5.6 We can reconsider Proposition 4.3 in the light o Proposition 5.5. For a given morphism F = (, ): in rr(), the universal property o the strong homotopy kernel gives a canonical comparison rom the kernel o F in rr() to the strong homotopy kernel o F in rr(). Explicitly, the canonical comparison is given by 15 Ker() k K() Ker( ) k, Now, Proposition 4.3 says that the canonical comparison is a weak equivalence. More precisely, the canonical comparison induces two morphisms K(k ): Ker(K()) Ker() and C(k ): Cok(K()) Cok() and we have that - K(k ) always is an isomorphism or a general argument o interchange o limits, - i and K() are proper and i is a regular epimorphism, then C(k ) also is an isomorphism. 6. The complete snail lemma From [3], Lemma 4.5.1, recall the ollowing act (which deines the homology o a complex). Proposition 6.1 Let be a pointed regular protomodular category. Consider the ollowing commutative diagram in, with a b = and a: X Y proper. a X Y a k b Ker(b) c a c a Cok(a ) Ker(b ) b Z b Cok(a) k b = k b c a. Mo- There exists a unique morphism i: Cok(a ) Ker(b ) such that c a i k b reover, the morphism i is a isomorphism.

16 16 In the rest o this section is abelian. Consider a commutative diagram in together with its actorization, through the pullback, and its actorization [, ] through the pushout +, as in the ollowing diagram,, Ker[, ] c, k [,] Cok, +, [,] Lemma 6.2 (With the previous notation.) There exists a unique morphism i: Cok, Ker[, ] such that c, i k [,] =. Moreover, the morphism i is an isomorphism. Proo. pply Proposition 6.1 to the complex : + [ :] where : and [ : ] are the actorizations o the pairs (, ) and (, ) through the product and the coproduct +, and the unlabelled isomorphism is given by ( ) id : id + Corollary 6.3 Consider a morphism F = (, ): in rr() and construct the commutative diagram id +,,, id [,]

17 17 1. The let-hand part is the strong homotopy kernel o F in rr(). 2. The right-hand part is the strong homotopy cokernel o F in rr(). 3. There is an exact sequence Ker, Ker() Ker() H(F) Cok() Cok() Cok[, ] where H(F), the homology o F, stands or the object Ker[, ] Cok,. Proo. Point 1 is just Proposition 5.5, and point 2 ollows rom point 1 by duality. s ar as point 3 is concerned, rom the snail lemma and its dual we get two exact sequences that we can past together thanks to Lemma 6.2. Remark Since the category is abelian, rr() is a 2-category, and it is easy to adapt the proo o Proposition 5.5 to show that the strong homotopy kernel o F is also a bikernel o F in the sense o [1]. Dually, the strong homotopy cokernel o F is also a bicokernel. 2. Moreover, rr() is biequivalent to the 2-category Grpd() o internal groupoids, internal unctors and internal natural transormations in. From this point o view, the exact sequence o Corollary 6.3 is a kind o π 1 -π -sequence associated to F, seen as an internal unctor. This is the approach adopted in [1], where non protomodular versions o the snail and o the snake lemma are discussed, in order to get an internalization o the classical exact sequence associated to a ibration o groupoids due to R. rown (see [5], or [6] or the non ibrational version). Reerences [1] J. énabou, Some remarks on 2-categorical agebra, ulletin de la Société Mathématique de elgique 41 (1989) [2] D. ourn, 3 3 lemma and protomodularity, Journal o lgebra 236 (21) [3] D. ourn, F. orceux, Mal cev, Protomodular, Homological and Semi-abelian categories, Kluwer cademic Publishers 24. [4] D. ourn, M. Gran, Regular, protomodular and abelian categories. In : Categorical Foundations, M.C. Pedicchio and W. Tholen Editors, Cambridge University Press (24) [5] R. rown, Fibrations o groupoids, Journal o lgebra 15 (197) [6] J. Duskin, R. Kieboom, E.M. Vitale, Morphisms o 2-groupoids and lowdimensional cohomology o crossed modules, Fields Institute Communication Series 43 (24) [7] M. Grandis, Simplicial homotopical algebras and satellites, pplied Categorical Structures 5 (1997)

18 [8] Z. Janelidze, Subtractive categories, pplied Categorical Structures 13 (25) [9] Z. Janelidze, E.M. Vitale, Snail lemma in a pointed regular category (in preparation). [1] S. Mantovani, G. Metere, E.M. Vitale, The snail lemma or internal groupoids (in preparation). Institut de recherche en mathématique et physique Université catholique de Louvain Chemin du Cyclotron Louvain-la-Neuve, elgique enrico.vitale@uclouvain.be 18

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