3 3 Lemma and Protomodularity

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1 Ž. Journal o Algebra 236, doi: jabr , available online at on 3 3 Lemma and Protomodularity Dominique Bourn Uniersite du Littoral, 220 a. de l Uniersite, BP 5526, Dunerque Cedex, France bourn@lmpa.univ-littoral.r Communicated by Walter Feit Received March 6, 2000 The classical 3 3 lemma and snae lemma, valid in any abelian category, still hold in any quasi-pointed Ž the map 0 1 is a mono., regular, and protomodular category. Some applications are given, in this abstract context, concerning the denormalization o ernel maps and the normalization o internals groupoids Ži.e., associated crossed modules Academic Press Key Words: short exact sequence; short ive lemma; 3 3 lemma; abelian category; regular and protomodular category. INTRODUCTION The notion o exact sequence has an intrinsic meaning in any pointed, regular, and protomodular category 2, among the examples o which there are the category o groups, rings, Lie algebras, Jordan algebras, any variety o -groups, any abelian category o course, the dual o the category o pointed sets, and more generally, any category o internal groups or internal rings in a let exact and regular category, any dual o the category o pointed objects in a topos, see also 10 or the notion o semi-abelian category. Thereore, the question naturally arises whether the classical results concerning the exact sequences still hold in this ind o category, which would put it as a good simple abstract setting or homological algebra in a nonabelian context. We show here that the 3 3 lemma and the snae lemma actually hold in it, and even in a slightly larger context, namely, that o quasi-pointed Žthe map 0 1 is no more an iso, but only a mono., regular, and protomodular category, deined here as the sequential categories. This allows us to integrate as examples any ibre $35.00 Copyright 2001 by Academic Press All rights o reproduction in any orm reserved. 778

2 3 3 LEMMA AND PROTOMODULARITY 779 Grd o the iltration Ž. 0: Grd associating with each internal groupoid its object o objects, when the basic category is let exact and regular. Some applications o these results are given about the denormalization o the ernel maps which gives rise to a characterization o some ernel equivalences associated with a morphism and about the normalization o the internal groupoids which gives rise to a characterization o the internal crossed modules associated with the connected internal groupoids. For the sae o brevity, we voluntarily restricted to what we could call the passive aspect o the 3 3 lemma, meaning by that the situation where all the morphisms are explicitly given. There is clearly some active versions o it, when it is possible to create some morphisms rom only a part o the 3 3 diagram. This would obviously be the case when the notion o normal monomorphism is clearly conceptually distinguished rom that o the ernel map. In this sense, this article is quite complementary to 4, where it is shown precisely that the notion o normal monomorphism also has an intrinsic meaning in any protomodular category, without any right exactness condition Žsee also. 5. The coordination between the two articles is rightly realized in the context o Barr exact categories 1, i.e., regular categories in which every equivalence relation is eective, that is the ernel equivalence o some map. The article is organized along the ollowing line: Ž. 1 Quasi-pointed categories. Ž. 2 Protomodular categories. Ž. 3 Sequentiable categories. Ž. 4 The 3 3 lemma. Ž. 5 The snae lemma. Ž. 6 Some applications. 1. QUASI-POINTED CATEGORIES We call quasi-pointed a let exact category with an initial object such that the map 0 1 is a monomorphism. This implies that, given an object in, there is at most one map 0 and that the ernel equivalence o this map is the same as the ernel equivalence o the terminal map 1, namely, the coarse equivalence gr.

3 780 DOMINIQUE BOURN Ž. A map will be said to be triial or null when it actors in a unique way through 0. The ernel o any map : Y is then deined by the ollowing pullbac: K er 0 Y Y The coernel o the map : Y is then any map q: Y Q which universally trivializes. This implies that is above 0, and deines this coernel as the pushout along o the map 0. When it exists, we shall denote it by coer and its codomain by Coer. The coernel o any map, when it exists, is a regular epimorphism, since in any category with pullbacs: Ž. 1 any map 0, being split, is a regular epimorphism and Ž. 2 the regular epimorphisms Ž being the quotient o their ernel pairs. are stable by pushouts whenever they exist. It is not the case in any quasi-pointed category that a regular epimorphism is the coernel o its ernel. Consider the category Sets o pointed sets, or instance. EAMPLE. Suppose is a let exact category and denote Grd the category o internal groupoids in. Let Ž. 0: Grd the unctor associating with each groupoid its object o objects. It is a ibration. Clearly Grd1, the ibre above 1, is just Gp the category o internal groups in. Any ibre Grd above any object is thus the category o internal groupoids with ixed objects. It is quasi-pointed by the ernel equivalence dis o 1 :. The ollowing result is classical in any quasi-pointed category : LEMMA 1. the map : Gien a commutatie diagram where the map is the ernel o K Y u w K Y Ž. 1 Suppose w is a monomorphism, then is the ernel o i and only i the let-hand side square is a pullbac. Ž. 2 Suppose the right-hand side square is a pullbac, then is the ernel o i and only i u is an isomorphism.

4 3 3 LEMMA AND PROTOMODULARITY PROTOMODULAR CATEGORIES We mentioned that in general coernels and regular epimorphisms do not coincide. This distinction will disappear in the ollowing context. We denote by Pt the category whose objects are the split epimorphisms in with a given splitting and morphisms the commutative squares between these data. We denote by :Ptthe unctor associating its codomain with any split epimorphism. As soon as has pullbacs, the unctor is a ibration which is called the ibration o pointed objects. The category is said to be protomodular 2 when has its change o base unctors conservative; i.e., relecting isomorphisms: when an arrow is mapped onto an isomorphism, it is an isomorphism. When is pointed, this condition is equivalent to the split short ive lemma, which maes the category Gp o groups the leading example o this notion. EAMPLE. When is let exact, then any ibre Grd above an object shares with the ibre Grd1 Gp the property o being protomodular. Any dual o an elementary topos is protomodular. Remar. In any quasi-pointed protomodular category, a map is a monomorphism i and only i its ernel is 0. More generally, pullbacs relect monomorphisms, see 2. As soon as a unctor F: preserves pullbacs and is conservative, protomodular implies protomodular. Accordingly, any ibre Pt o above an object is protomodular, as well as any slice category. The protomodularity condition is equivalent to the ollowing one: given a pullbac o split epimorphisms, then the pair Žu, s. is jointly strongly epic: u U U d s d s V It ollows rom that: PROPOSITION 2. In a quasi-pointed protomodular category, a map is a regular epi i and only i it is the coernel o its ernel. Proo. Given a map : Y, let us consider the ollowing diagram: V K K p p p p K 0 er Y Y

5 782 DOMINIQUE BOURN The map K 0, being split, is the quotient o its ernel equivalence. Let s denote the diagonal, the pair Ž, s. 0 Y 0 is then jointly strongly epic. Now, a map g: Z coequalizes the pair Ž p 0, p1. i and only i it coequalizes the pairs Ž p., p.. and Ž p.s, p.s In other words, i and only i g.er coequalizes the pair Ž p, p. 0 1, i.e., i and only i g.er actors through 0. Consequently, is a regular epi i and only i it is the coernel o its ernel. Whence, the ollowing deinition 2 : DEFINITION 3. Given a quasi-pointed protomodular category, a short exact sequence is a trivial sequence Ž i.e., with the composite trivial. such that is the ernel o and is the coernel o. We shall picture it in K Y. 3. SEQUENTIABLE CATEGORIES DEFINITION 4. We shall call sequentiable a category which is quasipointed, regular, and protomodular. This means 1 that, moreover, every eective equivalence relation Ži.e., ernel equivalence o some map. has a quotient and that regular epimorphisms are stable by pullbac. Remar. In this protomodular context, any ernel map has a coernel and the classical epi-mono actorization o a map : Y is obtained in the ollowing way: tae its ernel : K and then tae the coernel q: Q o. The actorization Q Y is a monomorphism. EAMPLE. When is regular, then any ibre Grd above an object is still regular. Consequently, when is let exact and regular, then any ibre Gr above an object is sequentiable. When is regular and protomodular, any ibre Pt is sequentiable. When is sequentiable, any slice category is sequentiable. The dual o any elementary topos being regular and protomodular, the ibres op Pt Ž. Pt op are sequentiable. Let us recall 2 that, in the presence o regularity, protomodularity is equivalent to the ollowing condition we shall need later on: PROPOSITION 5. Suppose the category is let exact and regular, it is protomodular i and only i the ollowing condition holds: gien a commuta-

6 3 3 LEMMA AND PROTOMODULARITY 783 tie diagram with the middle ertical map a regular epimorphism: i the let-hand side and the total rectangle are pullbacs, then the right-hand square is a pullbac. This gives a nice way o checing when a relexive graph is a ernel equivalence: COROLLARY 6. Gien an augmented relexie graph, it is the ernel equialence o its augmentation, i and only i the map d.er d is er. 1 0 Proo. Let us consider the ollowing augmented relexive graph: Y G d0 d 1 Now, consider the ollowing commutative diagram: er d0 d1 Kd G 0 d0 0 Y The middle vertical map is a regular epimorphism as being split. Then, apply the previous proposition. More generally, we get the short ive lemma in ull generality through the ollowing ind o converse o Lemma 1. PROPOSITION 7. Gien, in any sequentiable category, a commutatie diagram where is the ernel o and the upper row exact: K Y u w K Y Ž. 1 i the let-hand side square is a pullbac, then w is a mono. Ž. 2 i u is an isomorphism, then the right-hand side square is a pullbac. Ž. 3 i u and w are isomorphisms, then is an isomorphism Žshort ie lemma..

7 784 DOMINIQUE BOURN Proo. Ž. 1 The let-hand side square being a pullbac, is the ernel o. w.. The upper row is exact, thus w. is the epi-mono actorization o the map., and w is a mono. Ž. 2 Now, consider the ollowing commutative diagram: K 0 Y Y The middle vertical map is regular, the let-hand side square is a pullbac. But y..u and u is an isomorphism. Consequently, the total rectangle is also a pullbac. Then the right-hand side square is a pullbac. Ž. 3 Furthermore, when w is an iso, the map is itsel an iso. Our aim now is to prove that the 3 3 lemma holds in ull generality in any sequentiable category. For that, we need two preliminary results. PROPOSITION 8. w Gien a morphism o the preious ind: K Y u w K Y suppose w is an isomorphism, then u is a regular epimorphism i and only i is a regular epimorphism. Proo. The map w being an iso, the let-hand side square, ollowing Lemma 1, is a pullbac. So, when is a regular epi, u is a regular epi. Conversely, let us denote : Q the coernel o the ernel o, and is the monomorphism such that.. The map u being a regular epi, there is a actorization : K Q such that.u. and.. Now, let us set.. Then, w... The map w being an iso and being a regular epi, the map is a regular epi. On the other hand, being a mono, the ollowing square is a pullbac: K Q 1 K K Thus, according to Lemma 1, the map is the ernel o. Now, the short ive lemma implies that is an iso and. a regular epi.

8 3 3 LEMMA AND PROTOMODULARITY 785 More generally, starting rom a diagram o the previous ind, we can construct the ollowing diagram Ž., where the lower right-hand side square is a pullbac, K Y 1 u 1 Y K Z g Y h 1 K 2 w K Y the map g is then a regular epi since is so, and the middle row is exact. Moreover, the upper let-hand side square is a pullbac. COROLLARY 9. Consider any commutatie diagram as aboe: K Y u w K Y Ž. 1 when w and u are regular epimorphisms, then is a regular epimorphism. Ž. 2 When w is a monomorphism, then u is a mono i and only i is a mono. Proo. Straightorward considering the decomposition Ž. and the previous results. COROLLARY 10. Ž. 1 When u is a regular epimorphism, then the restriction Ž K.: K K w o the map to the ernels is a regular epimorphism. Ž. 2 When and w are split epimorphisms and the right-hand square commutes with the splittings, then the restriction KŽ.: K K w o the map to the ernels and the extension : Y w o the map to the ernel equialences o and w are necessarily regular epimorphisms. Proo. Ž. 1 The map u being a regular epi, such is the map 1. But the ollowing square is a pullbac, where : Kw Z is the unique map which is the ernel o and such that g. er w. Consequently, KŽ. 2

9 786 DOMINIQUE BOURN is a regular epi: KŽ. K Kw er 1 Z Ž. 2 When the right-hand square commutes with the splittings, the map u is split and then a regular epi, consequently the map KŽ.: K Kw is a regular epi. Now, consider the ollowing diagram: p 0 er p 0 K p 0 p 1 K Ž. w Y Y Y w K p er p 0 0 p 1 p0 then, according to Corollary 6, K p K and KŽ. KŽ. 0. Now, Ž K. and are regular epimorphisms, and, according to Corollary 9, the map is a regular epi. As an immediate consequence, we get that the regular epimorphisms in the category Gr d o internal groupoids in a sequentiable category are those internal unctors whose underlying morphisms o relexive graphs are Ž componentwise. regular epimorphisms. A protomodular category being always Mal cev Žsee. 4, this could have been indirectly derived rom 8, but in a less limpid way. We shall also need the ollowing technical result: PROPOSITION 11. Suppose we hae a commutatie diagram with the ernel o, and the lower row null: K Y u w K Y The preious decomposition Ž. is still possible ia the right-hand side lower pullbac. Suppose the map 1 is a regular epimorphism and the upper let-hand side square is a pullbac, then the map is the ernel o.

10 3 3 LEMMA AND PROTOMODULARITY 787 Proo. From the act that the upper let-hand side square is a pullbac, we can derive that the map h is a mono since is a mono and the category is protomodular. Now, let us tae a: A the ernel o. The lower right-hand side square being a pullbac, there is a unique map a : A Z which is the ernel o g and satisies.a 2 a. Consequently, there is a unique map : K A such that a. h, and h being a mono, this is itsel a mono. According to Lemma 1, the ollowing square is a pullbac: K. u 1 A a and thereore.u is a regular epi since it is the case or 1, which implies that is itsel a regular epi. Thus, is an isomorphism. Now, a..a..h and is the ernel o. 2 2 Z 4. THE 3 3 LEMMA THEOREM 12. Suppose gien the ollowing commutatie diagram in a sequentiable category, with the three rows exact: K Y u w K u w Y K Y Suppose the middle column is triial. I two among the three columns are exact, then the third one is exact. Proo. The map being a regular epi and the map being a mono, the nullity o the middle column implies the nullity o the two others. On the other hand, let us introduce the previous decomposition Ž.. Its middle row is still exact, and its upper let-hand side square is a pullbac. Ž. 1 Suppose the two last columns are exact. The map w being a mono, the upper let-hand square is a pullbac. The map being the ernel o, the map u is then the ernel o..u. But is a mono, thereore u is the ernel o u.

11 788 DOMINIQUE BOURN On the other hand, u is a regular epi since is a regular epi, according to Corollary 10. Ž. 2 Suppose the two extremal columns are exact. The last column being exact, the map w is a regular epi. The irst column being exact, the map u is a regular epi, and ollowing Corollary 9, such is. To show that is the ernel o, it is enough to prove that the ollowing square is a pullbac, 1 Y t Z where the map t: Y Z is the ernel o consider the ollowing diagram Ž.: 2 and satisies g.t w. Now, K 1 0 Y Z The let-hand square is a pullbac since the irst row is exact, the middle vertical map is a regular epi, and the total rectangle is equal to the ollowing one: t u K K u 1 0 K Z But this total rectangle is a pullbac, as made o two pullbacs, the let-hand side one meaning that the irst column is exact. Consequently, the right-hand square in the diagram Ž. is a pullbac. h Ž. 3 Suppose the two irst columns are exact. Then, is a regular epi. Thus, w.. is a regular epi, and such is w. On the other hand, u is a regular epi and, according to Proposition 8, the map 1 is a regular epi. Moreover, the ollowing square is a pullbac, exactly or the same reasons as previously in 2: 1 Y t Z

12 3 3 LEMMA AND PROTOMODULARITY 789 Now applying Proposition 11 to the diagram determined by the two last columns, the map w is the ernel o w. Actually, a careul analysis o the previous proo shows that: PROPOSITION 13. Let us consider a 3 3 diagram o the preious ind with the three rows exact and the middle column null. As soon as the irst column is exact, is the ernel o i and only i w is the ernel o w. 5. THE SNAKE LEMMA We shall call proper any map u whose monomorphic part o its epi-mono actorization is a ernel map. PROPOSITION 14. Suppose gien a morphism o exact sequences and that the coernel o the map u exists, K Y u w K Y there is a connecting morphism d: Kw Coeru such that d.kž. is triial. Moreoer, when u is proper, this triial sequence is exact at K w. When the coernel o the map exists, the map CoerŽ..d is triial. Moreoer, when is proper, then this triial sequence is exact at Coeru. When the coernel o the map w exists, and the maps and w are proper, then the sequence CoerŽ..CoerŽ. is exact at Coer. Proo. Let us denote er w: Kw Y the ernel o w and let us Ž denote K.: K Kw the restriction o to the ernels. Now, let us consider the ollowing pullbac: h H Kw Y er w

13 790 DOMINIQUE BOURN Then, h is the ernel o the map w... Thus, there is a map : H K such that the ollowing square is a pullbac: H K h On the other hand, there is a unique map : K H which is the ernel o the regular epimorphism and such that h.. The map is then the coernel o. Moreover,. u. Thus, i coer u denotes the coernel o u, then coer u.. coer u.u 0. Whence a unique map d: Kw Coeru such that d. coer u.. Now, because o the second mentioned pullbac, there is a map : K H which is the ernel o and such that h. er. It ollows rom that:. KŽ.. Thereore, d.kž. d.. coer u.. 0. Now, let u u.u be the epi-mono actorization o u. I u is proper, apply Corollary 10 to the ollowing morphism o exact sequences, K H Kw u d Q Coeru u coer u and the actorization K Ker Kd is a regular epimorphism. Thus, the sequence d.kž. is exact at Kw. Suppose now coer does exist and denote CoerŽ. the extension o to the coernels. Then, CoerŽ..d is trivial since: CoerŽ..d. CoerŽ..coer u. coer.. coer..h 0. Let. be the epi-mono actorization o the map and let us suppose proper. Then, is the ernel o coer. Let be the ernel o CoerŽ. and consider the ollowing square: K coer u coer Coeru CoerŽ. Coer I is the pullbac o along, then it is also the pullbac o along coer u. Now, consider the ollowing diagram where is such that. d.

14 3 3 LEMMA AND PROTOMODULARITY 791 Then, h H Kw V S T j K Coeru coer u The map j is a regular epi since the lower right-hand side square is a pullbac. The map such that. is a pullbac o since the let-hand side rectangle is a pullbac, and thus it is a regular epi. Consequently, the map is a regular epi and the trivial sequence CoerŽ..d is exact at Coeru. Let w be proper and let w w.w be its epi-mono actorization. Then, w is the ernel o coer w and the actorization such that w.. is a regular epi. Let : J Coer denote the ernel o CoerŽ., then thans to Corollary 10, the actorization : K J is a regular epi since it is the case or the map which satisies. w.. Now, i denotes the actorization o CoerŽ. through, it is a regular epi since.coer u. 6. SOME APPLICATIONS The ernel o a map is classically considered as the normalization o its ernel equivalence. We are somehow now going to denormalize some o our previous results on ernel maps and, conversely, to normalize some aspects o internal groupoids. PROPOSITION 15. Let us consider the ollowing diagram o augmented relexie graphs, where the upper graph is the ernel equialence o, and the ertical maps are regular epimorphisms: p 0 Y p 1 w u Y d 0 G d 1

15 792 DOMINIQUE BOURN Ž. 1 The lower graph is the ernel equialence o when the ernel extension o this diagram is exact, i.e., produces a ernel pair with its coernel: KŽ. KŽ p 0. Kw K Ku KŽ p 1. Ž. 2 When urthermore is a regular epi, then the lower graph is the ernel equialence o i and only i the ernel extension o this diagram is exact. Ž. 3 When the preious diagram determines a morphism o split augmented relexie graphs, then the lower graph is necessarily the ernel pair o. Ž. Proo. 1 Now consider the ollowing diagram: p 0 er p 0 Y K p 0 p 1 w u KŽ u. d 0 Y G Kd 0 d 1 er d 0 then, according to the 3 3 lemma, the ollowing sequence is exact: K Ž er u. KŽ u. K KŽ p. K p Kd. Then, consider the ollowing one: K Ž er u. KŽ u. K KŽ p. K p Kd KŽ p 1.er p0. p 1.er p0 d 1.er d0 K er KŽ. Kw er w Y Y Suppose the ernel extension is exact, then the irst column is exact and, according to Proposition 13, the last column is let exact since the central one is so. Thereore, the lower graph is the ernel equivalence o, according to Corollary 6. Ž. 2 I is a regular epi, then the central column is exact, and thereore the third one is exact i and only i the irst one is exact. This same Corollary 6 now ends the proo. w

16 3 3 LEMMA AND PROTOMODULARITY 793 Ž. 3 There is one circumstance where the previous conditions 1 and 2 are automatically ulilled, it is when the augmented relexive graphs are split; i.e., when there are extra maps s: Y and s : G such that 1 s. d.s, s.s s.s, and d.s 1, and the same ind o maps at the level o. Indeed, the ernel extension o our diagram produces a split augmented relexive graph. But clearly its relexive graph part is jointly monic, i.e., is a relexive relation. Now, a relexive relation in a protomodular category is always an equivalence relation since a protomodular category is Mal cev 4. This equivalence relation is, as such, an internal groupoid in and the splitting o the underlying graph produces a section o the actorization map: KŽ p., KŽ p.: Ku K K Žsee 3, Proposition K W which is itsel a mono. Consequently, this actorization is an isomorphism and our ernel extension is exact. Conversely, we are now going to investigate the normalization o the internal groupoids. It is well nown that, in the category Gp o abstract groups, the notion o the internal groupoid is equivalent to the notion o the crossed module, see or instance 6, where a crossed module is given by a homomorphism h: H and a let action o the group on the group H such that h is a homomorphism o let actions Ž being endowed with the action on. Ž. 1 itsel by conjugation which moreover satisies: h y z y. z. y. The homomorphism underlying the crossed module associated with an internal groupoid is actually given in the category Gp by the ollowing very general construction which maes sense, in any quasi-pointed let exact, category. Given an internal groupoid 1, let us consider the ollowing diagram, d p0 p1 p0 p1 d0 d1 Kd 0 er d 0 d 0 1 d 0 Kd Kd d 0 0 where the map d 2, here, represents the operation which, in the set theoretical context, would be deined or every pair o arrows Ž,. with Ž. 1 same domain by the ormula d,.. 2 DEFINITION 16. We shall reer to the map h d.er d as the normal- 1 0 ization o the internal groupoid. 1

17 794 DOMINIQUE BOURN When Gp, the homomorphism underlying the crossed module associated with 1 is this normalization. Now, in any case, all the commutative squares o this diagram are pullbacs. As a consequence, the map s.er h: Kh Kd Kd is the ernel o d 2., which will allow us to characterize the normalization o the connected internal groupoids. PROPOSITION 17. Gien a sequentiable category, then a regular epimorphism h: H is the normalization o a groupoid i and only i the map s.er h: Kh 0 H H is a ernel map. When moreoer is pointed, these inds o regular epimorphisms characterize the connected groupoids. Proo. A connected groupoid is such that the map d, d 0 1 : is a regular epimorphism. When the basic category is pointed, the normalization o a groupoid 1 can be given by the ollowing pullbac as well: H 1 h0 1 d, d 0 s Consequently, when the groupoid is connected, then its normalization is a regular epi. Conversely, let h be a regular epimorphism such that the map s 0.er h: Kh H H is a ernel map. Then, consider 1 the codomain o the coernel q o s.er h. This produces a relexive graph d, d : 1 on such that the commutative squares in the ollowing diagrams are pullbacs, according to Proposition 7: H H q 1 p p 0 1 d0 d1 H h On the one hand, this graph is underlying a groupoid. Indeed the actorization q: H H H d 1 0 is a regular epimorphism since it is produced by the extension o the previous pullbac with the irst projections to the ernel equivalences o p0and d 0. Consequently, the groupoid 1 appears to be the coernel in the category Grd o the internal unctor: dis Kh gr H, where dis Z is the ernel equivalence o 1 Z: Z Z and gr Z is the ernel equivalence o the terminal map Z 1. Moreover, d, d.q h h which is a regular epi, thereore d, d is a regular epi and the groupoid is connected. 1

18 3 3 LEMMA AND PROTOMODULARITY 795 On the other hand, the ernel o d being q.s we have: d.er d d.q.s h. p.s h It is certainly worth noticing that, when Gp, the previous regular epimorphisms are precisely those which have central ernels, i.e., those which correspond to central extensions 12. In the wider context o commutative algebra, a number o examples o such internal crossed modules has been studied in 13. See also 11, 7, 9. REFERENCES 1. M. Barr, Exact Categories, Springer Lecture Notes in Mathematics, Vol. 236, pp. 1120, Springer-Verlag, BerlinNew Yor, D. Bourn, Normalization Equivalence, Kernel Equivalence and Aine Categories, Springer Lecture Notes in Mathematics, Vol. 1488, pp. 4362, Springer-Verlag, Berlin New Yor, D. Bourn, Polyhedral monadicity o n-groupoids and standardized adjunction, J. Pure Appl. Algebra 99 Ž 1995., D. Bourn, Normal subobjects and abelian objects in protomodular categories, J. Algebra 228 Ž 2000., D. Bourn, Normal unctors and strong protomodularity, Theory Appl. Categories 7 Ž 2000., R. Brown and C. Spencer, Double groupoids and crossed modules, Cahiers Topologie Geom. Dierentielle Categoriques 17 Ž 1976., G. J. Ellis, Homotopical aspects o Lie algebras, J. Austral. Math. Soc. Ser. A 54 Ž 1993., M. Gran, Central extensions or internal groupoids in Maltsev categories, These, ` Louvain la Neuve, A. R. Grandjean and M. Ladra, Crossed modules and homology, J. Pure Appl. Algebra 95 Ž 1994., G. Janelidze, L. Mari, and W. Tholen, Semi-abelian categories, preprint, A. S.-T. Lue, Cohomology o algebras relative to a variety, Math. Z. 121 Ž 1971., S. MacLane, Homology, Springer-Verlag, BerlinNew Yor, T. Porter, Some categorical results in the theory o crossed modules in commutative algebras, J. Algebra 109 Ž 1987.,