Chain complexes of symmetric categorical groups

Journal of Pure and Applied Algebra 196 (25) 279 312 www.elsevier.com/locate/jpaa Chain complexes of symmetric categorical groups A. del Río a, J. Martínez-Moreno b, E.M. Vitale c,,1 a Departamento de Álgebra, aculdad de Ciencias, Universidad de ranada 1871 ranada, Spain b Departamento de Matemáticas, Universidad de Jaén, 2371 Jaén, Spain c Département de Mathématiques, Université catholique de Louvain Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium Received 9 September 23; received in revised form 5 May 24 Communicated by I. Moerdijk Available online 12 October 24 Abstract We define the cohomology categorical groups of a complex of symmetric categorical groups, and we construct a long 2-exact sequence from an extension of complexes. As special cases, we obtain Ulbrich cohomology of Picard categories and the Hattori Villamayor Zelinsky sequences associated with a ring homomorphism. Applications to simplicial cohomology with coefficients in a symmetric categorical group, and to derivations of categorical groups are also discussed. 24 Elsevier B.V. All rights reserved. MSC: 185; 18D1; 18B4; 13D99; 183 1. Introduction In the late seventies, Villamayor and Zelinsky [18] and, independently, Hattori [9], discovered a long exact sequence connecting Amitsur cohomology groups of a commutative algebra with coefficients U (the group of units) and Pic (the Picard group). The search of a better understanding of the Hattori Villamayor Zelinsky sequence lead to a series of works by Takeuchi and Ulbrich culminating with a cohomology theory for Picard categories [13 17]. Corresponding author. E-mail addresses: adelrio@ugr.es (A. del Río), jmmoreno@ujaen.es (J. Martínez-Moreno), vitale@math.ucl.ac.be (E.M. Vitale). 1 Supported by NRS rant 1.5.116.1. 22-449/$ - see front matter 24 Elsevier B.V. All rights reserved. doi:1.116/j.jpaa.24.8.29

28 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 The aim of this work is to revisit the previous results using recent techniques developed in higher dimensional homological algebra. In fact, we will derive Hattori Villamayor Zelinsky sequence and Ulbrich cohomology as special instances of general results on the homology of symmetric categorical groups. The plan of the paper is as follows. Section 2: the kernel and the cokernel of a morphism between symmetric categorical groups have been studied in [1,19]. Here we refine these notions, introducing kernel and cokernel relative to a natural transformation, as in the following diagram Ker (,) Coker (,) Section 3: using relative kernels and cokernels, we define the cohomology categorical groups of a complex of symmetric categorical groups. As for abelian groups, there are two possible definitions, giving equivalent cohomology categorical groups. Section 4: an extension of (symmetric) categorical groups is a diagram which is 2-exact in the sense of [1,19] and such that is faithful and is essentially surjective (see [1,12]). ollowing the lines of [11], we associate a long 2-exact sequence of cohomology categorical groups to any extension of complexes of symmetric categorical groups. Section 5: we specialize the previous result to get Ulbrich cohomology and Hattori Villamayor Zelinsky exact sequences. We discuss also simplicial cohomology with coefficients in a symmetric categorical group. Section 6: in [7], a six term 2-exact sequence involving the low-dimensional cohomology of a categorical group with coefficients in a symmetric -module is constructed. We obtain this sequence as a special case of the kernel-cokernel lemma for symmetric categorical groups, which is a special case of the long cohomology sequence obtained in Section 4. 2. Relative kernel and cokernel A (symmetric) categorical group is a (symmetric) monoidal groupoid in which each object is invertible, up to isomorphism, with respect to the tensor product. We write C for the 2-category of categorical groups, monoidal functors, and monoidal natural transformations (which always are natural isomorphisms); SC is the 2-category of symmetric categorical groups, monoidal functors compatible with the symmetry, and monoidal natural transformations. or basic facts on (symmetric) categorical groups, we refer to [1,19] and

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 281 the references therein. As far as notations are concerned, if is a (symmetric) categorical group, we write π () for the (abelian) group of its connected components, and π 1 () for the abelian group of automorphisms of the unit object, that is π 1 () = (I, I). If is a group, we write [] for the discrete categorical group having the elements of as objects. If is abelian, we write [1] for the categorical group with just one object and having the elements of as arrows (if is not abelian, [1] is just a groupoid). If X is an object of a categorical group, we denote by X a fixed dual of X. (Note: composition is always written diagrammatically.) 2.1. The relative kernel iven a morphism : A B in C, the notation for its kernel, see [1,19], is Ker e ε We consider now two composable morphisms and in C, such that the composite is naturally equivalent to the zero functor, and we construct the relative kernel as in the following diagram Ker (,) e (,) ε(,) (1) The relative kernel Ker(,) is in C (in SC if, and are in SC), and it can be described as follows: An object is a pair (A A, a: A I) such that the following diagram commutes (A) a I A I I An arrow f : (A, a) (A,a ) is an arrow f : A A such that the following diagram commutes A f A a I a

282 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 The faithful functor e (,) is defined by e (,) (A, a)=a, and the natural transformation ε (,) by ε (,) (A, a) = a. The natural transformation ε (,) is compatible with, in the sense that the following diagram commutes e (,) ε (,) e (,) can e (,) can The relative kernel is a bi-limit, in the sense that it satisfies the following universal property (and it is determined by this property, up to equivalence): given a diagram in C E ψ (2) with ψ compatible with, there is a factorization (E : K Ker(,), ψ : E e (,) E) in C of (E, ψ) through (e (,), ε (,) ), that is the following diagram commutes E' e (,) E' ε (,) E' ψ' can E ψ and, if (E, ψ ) is another factorization of (E, ψ) through (e (,), ε (,) ), then there is a unique 2-cell e : E E such that e E e. e. (,) (,) E. ψ ψ E e (,) commutes. The relative kernel is also a standard homotopy kernel, in the sense that it satisfies the following universal property (and it is determined by this property, up to isomorphism): in the situation of diagram (2), there is a unique E : K Ker(,) in C such that E e (,) = E and E ε (,) = ψ.

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 283 To prove the previous universal properties is a simple exercise. The next proposition expresses the kind of injectivity measured by the relative kernel (compare with the similar results stated in [1,19] for the usual kernel). Proposition 2.1. With the notations of diagram (1). 1. π 1 (Ker(, )) = if and only if is faithful; 2. π (Ker(, )) = if and only if is -full (this means full with respect to arrows g : A 1 A 2 such that (g) A2 = A1 ). Proof. (1) We know from [1,19] that π 1 (Ker )= if and only if is faithful. Moreover, the comparison between Ker(,) and Ker is full and faithful, so that it induces an isomorphism between π 1 (Ker( )) and π 1 (Ker ). (2) Let (A, a) be an object of Ker(,). The arrow a I : A I I is such that (a I ) I = A.If is -full, there exists α : A I such that (α) = a I. This means that a realizes an isomorphism between (A, a) and the unit object of Ker(,). Conversely, if g : A 1 A 2 is such that (g) A2 = A1, then the following is an object of Ker(,) (A 1 A 2,g 1 : (A 1 A 2 ) A 1 A 2 A 2 A 2 I). If π (Ker(, ))=, there is a morphism h : (A 1 A 1 2,g 1) (I, I ) in Ker(,). Now, if we call f : A 1 A 2 the following composition h 1 A 1 A 1 I A 1 A 2 A 2 I A 2 A 2 we have that (f)= g. A direct consequence of the universal property (as a bi-limit) of the relative kernel is the following cancellation property. Proposition 2.2. In the situation of diagram (1), consider the following diagram in C H Ker (,) α e (,) If α and ε (,) are compatible, then there is a unique 2-cell ᾱ : H such that ᾱ e (,) = α. Proof. This is because (H, α) and (,can : e (,) ) provide two factorizations of (,can: ) through the relative kernel.

284 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 To finish, let us observe that the usual kernel is a special case of the relative one. Indeed, given a morphism : A B in C, we can consider the canonical natural isomorphism can and the relative kernel Ker(,can) is nothing but the usual kernel Ker. In particular, can-full just means full. 2.2. The relative cokernel iven a morphism : B C in SC, the notation for its cokernel, see [1,19] is π P Coker The picture for the relative cokernel is the following one (everything is in SC): π (,) P (,) Coker (,) (3) The relative cokernel Coker(,)can be described as follows: objects are those of C; pre-arrows are pairs (B, f ) : X Y with B B and f : X B Y ; an arrow is a class of pre-arrows, two pre-arrows (B, f ), (B,f ) : X Y are equivalent if there is A A and a : B (A) B such that the following diagram commutes X f f B Y a 1 B Y I B Y A 1 1 (A B ) Y (A) B Y the essentially surjective functor P (,) and the natural transformation π (,) defined as for the usual cokernel.

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 285 Once again, the natural transformation π (,) is compatible with, in the sense that the following diagram commutes P (,) π (,) P (,) can P (,) can Like the relative kernel, the relative cokernel is both a bi-limit and a standard homotopy cokernel with respect to diagrams in SC of the following kind P ψ (4) where ψ is compatible with in the obvious sense. We leave to the reader to state the universal properties and the cancellation property for the relative cokernel. In the next proposition, we fix the kind of surjectivity measured by the relative cokernel. Proposition 2.3. With the notations of diagram (3). 1. π (Coker(, )) = if and only if is essentially surjective; 2. π 1 (Coker(, ))= if and only if is -full (this means that, given h : B 1 B 2, there is A A and g : B 1 A B 2 such that h = (g) ( A 1 B2 )). Proof. (1) rom [1,19], we know that π (Coker )= if and only if is essentially surjective. Moreover, the comparison between Coker and Coker(,)is full and essentially surjective, so that it induces an isomorphism between π (Coker ) and π (Coker(, )). (2) Let [B B, b: I B I] :I I be a morphism in Coker(,). The arrow b gives rise to an arrow h : I I B I B in C.If is -full, there is A A and g : I A B such that h = (g) ( A 1 B ). The pair (A, g) attests that the morphism [B,b] is equal to the identity on I in Coker(,). Conversely, let g : B 1 B 2 be a morphism in C. We get the following morphism in Coker(,) π (B 1 ) 1 P I B (h) π 1 B (B 2 ) 2 I If π 1 (Coker(, )) =, the previous morphism is equal to the identity. This means that there is A A and a : B 1 A B 2 such that (a) ( A 1) = h. The usual cokernel is a particular case of the relative cokernel. Indeed, given a morphism : B C in SC, its cokernel is the relative cokernel Coker(can, ) as in the following

286 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 diagram can Once again, can-full just means full. 2.3. 2-exactness and relative 2-exactness Let us recall from [1,19] the notion of 2-exactness. Consider a sequence (,,) in SC together with the canonical factorizations through the kernel and the cokernel Ker e Coker Ker P Coker We say that the sequence (,,) is 2-exact if the functor is full and essentially surjective on objects or, equivalently, if the functor is full and faithful. This is also equivalent to say that Coker (or Ker ) is equivalent to. Consider now the following diagram in SC α L M with α compatible with and compatible wit γ. By the universal property of the relative kernel Ker(,γ), we get a factorization (, ) of (, ) through (e (,γ), ε (,γ) ).Bythe cancellation property of e (,γ),wehavea2-cellᾱ as in the following diagram L α e (,γ) γ Ker (,γ) Coker (α, )

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 287 The dual construction gives rise to the following diagram M γ P (α,) Ker (,γ) Coker (α,) A direct calculation shows that Coker(ᾱ, ) Ker(, γ). We say that the sequence (L, α,,,,γ,m)is relative 2-exact if the functor is essentially surjective and ᾱ-full or, equivalently, if the functor is faithful and γ-full. This is also equivalent to say that Coker(ᾱ, ) (or Ker(, γ)) is equivalent to. Since the comparison Ker(,γ) Ker is full and faithful, 2-exactness always implies relative 2-exactness. To make clear the difference between 2-exactness and relative 2-exactness, let us consider two basic examples. Example 2.4. 1. Consider the following sequence in SC A B P Coker together with can : and π : P. It is always 2-exact in B. It is 2-exact in A if and only if is full and faithful. Moreover, it is relative 2-exact in A if and only if is faithful. Indeed, π (Ker(, π )) =, so that any functor is π -full. 2. Consider the following sequence in SC Ker e B C together with ε : e and can :. It is always 2-exact in B. It is 2-exact in C if and only if is full and essentially surjective. Moreover, it is relative 2-exact in C if and only if is essentially surjective. Indeed, π 1 (Coker(can, )) =, so that any functor is ε -full. 3. The cohomology of a complex rom [11,13], recall that a complex of symmetric categorical groups is a diagram in SC of the form A = A L A 1 L 1 A 2 L 2 Ln 1 A n L n A n+1 L n+1

288 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 together with a family of 2-cells {α n : L n L n+1 } n such that, for all n, the following diagram commutes L n+1 L n L n+1 L n 1 αn L n 1 α n 1 L n+1 L n+1 can can To define the nth cohomology categorical group of the complex A, we use the following part of the complex α n 2 n 2 L n 2 n 1 L n 1 n L n n+1 L n+1 n+2 α n 1 and we repeat the construction given in 2.3: by the universal property of the relative kernel Ker(L n, α n ), we get a factorization (L n 1, α n 1 ) of (L n 1, α n 1 ) through (e (Ln,α n ), ε (Ln,α n )). By the cancellation property of e (Ln,α n ),wehavea2-cellᾱ n 2 as in the following diagram α n L n 2 L n 1 L n n 2 n 1 n n+1 α n 2 α n 1 L n 1 e (Ln,αn) Ker (L n,α n ) Coker (α n 2,L n 1 ) Definition 3.1. With the previous notations, we define the nth cohomology categorical group of the complex A as the following relative cokernel H n (A ) = Coker(ᾱ n 2,L n 1 ). Note that, as in Section 2.3, there is a dual construction of H n (A ) starting with the relative cokernel Coker(α n 2,L n 1 ) and ending with a convenient relative kernel. The resulting categorical groups are equivalent. Note also that, to get H (A ) and H 1 (A ),we have to complete the complex A on the left with two zero-morphisms and two canonical 2-cells A L A 1..., can:, can : L

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 289 We give now an explicit description of H n (A ): an object of H n (A ) is an object of the relative kernel Ker(L n, α n ), that is a pair (A n A n,a n : L n (A n ) I) such that L n+1 (a n ) = α n (A n ); a pre-arrow (A n,a n ) (A n,a n ) is a pair (X n 1 A n 1,x n 1 : A n L n 1 (X n 1 ) A n ) such that the following diagram commutes L n (A n ) a n a n L n (x n 1 ) L n (L n 1 (X n 1 ) A n ) I L n (L n 1 (X n 1 )) L n (A n ) α n 1(X n 1 ) 1 L n (A n ) I L n (A n ) an arrow is a class of pre-arrows; two parallel pre-arrows (X n 1,x n 1 ), (X n 1,x n 1 ) : (A n,a n ) (A n,a n ) are equivalent if there is a pair (P n 2 A n 2,p n 2 : X n 1 L n 2 (P n 2 ) X n 1 ) such that the following diagram commutes A n x n 1 L n 1 (X n 1 ) A n x n 1 L n 1 (X n 1 ) A n L n 1 (p n 2 ) 1 L n 1 (L n 2 (P n 2 ) X n 1 ) A n I L n 1 (X n 1 ) A n L n 1 (L n 2 (P n 2 )) L n 1 (X n 1 ) A n α n 2 (P n 2 ) 1 1 Remark 3.2. rom the previous description, it is evident that π (H n (A )) π 1 (H n+1 (A )). This will be useful in Section 5 to make some proofs shorter.

29 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 Let us look now at the functoriality of H n. A morphism : A B of complexes in SC is pictured in the following diagram:... n 1 L n 1 α n 1 n L n n+1... n 1... n 1 λ n 1 M n 1 λ n n n+1 n β n 1 M n n+1... where the family of 2-cells {λ n : L n n+1 n M n } n makes commutative the following diagram: L n 1 λ n λ n 1 M n L n 1 L n n+1 L n 1 n M n n 1 M n 1 M n α n 1 n+1 n+1 n 1 can can n 1 β n 1 Such a morphism induces, for each n, a morphism of symmetric categorical groups H n ( ) : H n (A ) H n (B ). Its existence follows from the universal property of the relative kernels and cokernels involved. It can be described explicitly: given an object (A n A n,a n : L n (A n ) I) in H n (A ),wehave H n ( )(A n,a n ) = ( n (A n ) B n, λ 1 n (A n) n+1 (a n ) : M n ( n (A n )) n+1 (L n (A n )) n+1 (I) I). The fact that ( n (A n ), λ 1 n (A n) n+1 (a n )) is an object of the relative kernel Ker(M n, β n ) depends on the condition on the family {λ n }. iven an arrow [X n 1 A n 1,x n 1 : A n L n 1 (X n 1 ) A n ]:(A n,a n ) (A n,a n ) in H n (A ),wehave H n ( )[X n 1,x n 1 ]=[ n 1 (X n 1 ) B n 1, n (x n 1 ) (λ n 1 (X n 1 ) 1) : n (A n ) n (L n 1 (X n 1 ) A n ) n (L n 1 (X n 1 )) n (A n ) M n 1 ( n 1 (X n 1 )) n (A n )].

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 291 4. The longcohomology sequence Recall that an extension of symmetric categorical groups is a diagram in SC which is 2-exact, is faithful and is essentially surjective (see [1,12]). Equivalently, an extension is a diagram in SC of the form can can which is relative 2-exact in A, B and C. (Indeed, 2-exactness and relative 2-exactness are equivalent conditions in B, because Ker(,can) Ker and can-full means full. Now, if the factorization of through Coker is full and faithful, the relative 2-exactness in A of (,can,,,)is equivalent to the relative 2-exactness in A of (,can,,π,p ), that is, by Example 2.4, to the faithfulness of. The argument for the essential surjectivity of is dual.) A morphism of extensions is pictured in the following diagram: L λ M μ N where the 2-cells make commutative the following diagram: N.μ M ' λ ' L ' ' N N L can can L. '

292 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 Definition 4.1. An extension of complexes in SC is a diagram where A B C are morphisms of complexes, and ={ n : n n } n is a family of 2-cells such that, for each n, n n n n n n is an extension of symmetric categorical groups and n n n n n n L n λ n M n μ n N n n+1 n+1 n+1 n+1 n+1 n+1 is a morphism of extensions. Theorem 4.2. Let be an extension of complexes of symmetric categorical groups. or each n, there is a morphism Δ n and three 2-cells H n ( ), Σ n and Ψ n making the following long sequence 2-exact

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 293 in each point H n ( ) ψ n H n ( ) Hn ( ) H n ( ) H n+1 ( ) H n ( ) H n Δ ( ) n H n+1 ( ) H n+1 ( ) Σ n Proof. We give the construction of the morphisms and 2-cells involved in the statement. As far as 2-exactness is concerned, we concentrate on the 2-exactness in H n (C ), which is the most delicate part of the proof. In fact, we give a first construction of Δ n and Σ n.we use these constructions to show that the factorization of H n ( ) through the kernel of Δ n is essentially surjective. Then we give a second construction of H n (C ), Δ n and Σ n, and we use them to show that the factorization of H n ( ) is full. Construction of H n ( ): given an object (A n A n,a n : L n (A n ) I) in H n (A ),if we apply H n ( ) and H n ( ) we obtain the following object of H n (C ): ( n ( n (A n )) C n, μ 1 n ( n(a n )) n+1 (λ 1 n (A n)) n+1 ( n+1 (a n )) : N n ( n ( n (A n ))) n+1 (M n ( n (A n ))) n+1 ( n+1 (L n (A n ))) n+1 ( n+1 (I)) I). Such an object is naturally isomorphic to (I C n,n n (I) I), which is the unit object in H n (C ), via the morphism H n ( ) =[I C n 1, n (A n ) : n ( n (A n )) I N n 1 (I) I]. irst construction of Δ n : let (C n Cn, c n : N n (C n ) I)be an object in Ker N n ; since n : B n C n is essentially surjective, there are B n B n and i : n (B n ) C n. Since (M n (B n ), μ n (B n ) N n (i) c n : n+1 (M n (B n )) N n ( n (B n )) N n (C n ) I) is an object of Ker n+1 and the factorization of n+1 : A n+1 B n+1 through Ker n+1 is an equivalence, there are A n+1 A n+1 and j : n+1 (A n+1 ) M n (B n ) such that n+1 (j) μ n (B n ) N n (i) c n = n+1 (A n+1 ). Now we need an arrow a n+1 : L n+1 (A n+1 ) I. Since the factorization n+2 of n+2 through Ker n+2 is an equivalence, it is enough to find an arrow n+2 (L n+1(a n+1 )) n+2 (I). This is given by λ n+1 (A n+1 ) M n+1 (j) β n (B n ) : n+2 (L n+1 (A n+1 )) I n+2 (I). inally, we put Δ n (C n,c n ) = (A n+1,a n+1 ). This is an object of H n+1 (A ): the condition L n+2 (a n+1 ) = α n+1 (A n+1 ) can be checked applying the faithful functor n+3. Consider now an arrow [Z n 1 C n 1,z n 1 : C n N n 1 (Z n 1 ) C n ]:(C n,c n ) (C n,c n )

294 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 in H n (C ). We look for an arrow [X n A n,x n : A n+1 L n (X n ) A n+1 ]:(A n+1,a n+1 ) (A n+1,a n+1 ) in H n+1 (A ). Since n 1 : B n 1 C n 1 is essentially surjective, there are Y n 1 B n 1 and l : n 1 (Y n 1 ) Z n 1. We get the following arrow in C n i z n 1 (N n 1 (l 1 ) 1) (μ 1 n 1 (Y n 1) 1) (1 (i ) 1 ) : n (B n ) n (M n 1 (Y n 1 )) n (B n ) n (M n 1 (Y n 1 ) B n ). Since the factorization of n through Coker n is an equivalence, we get the corresponding arrow in Coker n [X n A n,s : B n n (X n ) M n 1 (Y n 1 ) B n ]:B n M n 1 (Y n 1 ) B n. This allows us to construct an arrow n+1 (A n+1) n+1 (L n(x n ) A n+1 ) in Ker n+1 in the following way: n+1 (A n+1 ) j M n (B n ) M n(s) M n ( n (X n ) M n 1 (Y n 1 ) B n ) M n ( n (X n )) M n (M n 1 (Y n 1 )) M n (B n ) 1 β n 1 (Y n 1) 1 M n ( n (X n )) M n (B n ) λ 1 n (X n) (j ) 1 n+1 (L n (X n )) n+1 (A n+1 ) n+1(l n (X n ) A n+1 ). Since n+1 is an equivalence, we get a uniquely determined arrow x n : A n+1 L n (X n ) A n+1. inally, we put Δ n[z n 1,z n 1 ]=[X n,x n ]: the condition to be an arrow in H n+1 (A ) can be checked applying the faithful functor n+2. irst construction of Σ n : let (B n B n,b n : M n (B n ) I) be an object of H n (B );we put Σ n (B n,b n ) =[I A n, σ(b n,b n ) : A n+1 L n (I)],

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 295 where (A n+1,a n+1 ) = Δ n (H n ( )(B n,b n )) and σ(b n,b n ) corresponds to the arrow j b n : n+1 (A n+1) = n+1 (A n+1 ) I n+1 (L n (I)) = n+1 (L n(i)) of Ker n+1 via the equivalence n+1 : A n+1 Ker n+1. Indeed, the fact that [I,σ(B n,b n )] is an arrow in H n+1 (A ) can be checked applying the faithful functor n+2. 2-exactness in H n (C ): let us call Γ the factorization of H n ( ) through Ker Δ n.we are going to prove that Γ is essentially surjective. Let (C n C n,c n : N n (C n ) I),[ C n A n, c n : A n+1 L n ( C n )]: (A n+1,a n+1 ) = Δ n (C n,c n ) I be an object of Ker Δ n. Using the notations introduced in the first construction of Δ n,we construct the following object of H n (B ): ( n ( C n ) B n, τ = (λ 1 n ( C n ) 1) ( n+1( c n ) 1) ((j 1 ) 1) : M n ( n ( C n ) B n) M n ( n ( C n )) M n(b n ) M n (B n ) M n (B n ) I) and the needed isomorphism is given by Γ( n ( C n ) B n, τ) (C n,c n ), [ C n, c n ] [I C n 1, n ( C n ) i : n( n ( C n ) B n) n ( n ( C n )) n(b n ) C n ]. Second description of H n (C ): since ( n, n, n ) is an extension, C n is equivalent to the cokernel of n, and we get the following description of H n (C ). An object is a pair (B n B n, [A n+1 A n+1,a n+1 : M n (B n ) n+1 (A n+1 )]), where [A n+1,a n+1 ]:M n (B n ) I is an arrow in Coker n+1, such that there exists t n+2 : L n+1 (A n+1 ) I making commutative the following diagram M n+1 (M n (B n )) β n (B n ) M n+1 (a n+1 ) M n+1 ( n+1 (A n+1 )) λ 1 n+1 (A n+1) I n+2 (I) n+2 (L n+1 (A n+1 )) n+2 (t n+2 ) (note that such an arrow t n+2 is necessarily unique because n+2 is faithful). An arrow (B n, [A n+1,a n+1 ]) (B n, [A n+1,a n+1 ]) is a class of pairs (B n 1 B n 1, [A n A n,a n : B n n (A n ) M n 1 (B n 1 ) B n ]), where [A n,a n ]:B n M n 1 (B n 1 ) B n is an arrow in Coker n, such that there exists ā n : A n+1 L n (A n ) A n+1 making commutative the following diagram:

296 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 M n (B n ) an+1 M n (a n ) n+1 (A n+1 ) n+1 (a n ) n+1 (L n (A n ) A n+1 ) M n ( n (A n ) M n 1 (B n 1 ) B n ) M n ( n (A n )) M n (M n 1 (B n 1 )) M n (B n ) n+1 (L n (A n )) n+1 (A n+1 ) 1 a n+1 1 λ n (A n ) β n 1 (B n 1 ) 1 n+1 (A n (A n ) M n (B n ) (once again the arrow ā n is necessarily unique because n+1 is faithful). inally, two parallel pairs (B n 1, [A n,a n ]) and (B n 1 [A n,a n ]) are identified if there are B n 2 B n 2,A n 1 A n 1,a n 1 : B n 1 n 1 (A n 1 ) M n 2 (B n 2 ) B n 1 and ā n 1 : A n A n L n 1 (A n 1 ) such that the following compositions are equal B n a n n (A n ) M n 1 (B n 1 ) B n 1 M n 1(a n 1 ) 1 n (A n ) M n 1 ( n 1 (A n 1 ) M n 2 (B n 2 ) B n 1 ) B n 1 λ 1 n 1 (A n 1) β n 2 (B n 2 ) 1 n (A n ) n (L n 1 (A n 1 )) M n 1 (B n 1 ) B n n (A n L n 1 (A n 1 )) M n 1 (B n 1 ) B n a n B n n (A n ) M n 1(B n 1 ) B n n (ā n 1 ) 1 1 n (A n L n 1 (A n 1 )) M n 1 (B n 1 ) B n.

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 297 Second construction of Δ n : using the second description of H n (C ), we can define the functor Δ n : H n (C ) H n+1 (A ) on objects by Δ n (B n, [A n+1,a n+1 ]) = (A n+1,t n+2 : L n+1 (A n+1 ) I) and on arrows by Δ n [B n 1, [A n,a n ]]=[A n, ā n : A n+1 L n (A n ) A n+1 ]. Second description of H n ( ): we have to adapt the description of the functor H n ( ) : H n (B ) H n (C ) to the second description of H n (C ). The image of an object (B n B n,b n : M n (B n ) I) of H n (B ) is the object (B n B n, [I A n+1,b n : M n (B n ) I n+1 (I)]) (as arrow t n+2 one takes the canonical isomorphism L n+1 (I) I) and the image of an arrow [Y n 1 B n 1,y n 1 : B n M n 1 (Y n 1 ) B n ] : (B n,b n ) (B n,b n ) of H n (B ) is the arrow [Y n 1, [I A n,y n 1 : B n M n 1 (Y n 1 ) B n n(i) M n 1 (Y n 1 ) B n ]]. Second construction of Σ n : using the second description of the functors Δ n and H n ( ), the 2-cell Σ n is the identity 2-cell. 2-exactness in H n (C ): we are going to prove that Γ : H n (B ) Ker Δ n is full. or this, observe that an object in Ker Δ n is a pair (B n, [A n+1,a n+1 ]) H n (C ), [X n,x n ] : (A n+1,t n+2 ) I H n+1 (A ) and an arrow in Ker Δ n is an arrow [B n 1, [A n,a n ]] in H n (C ) (with its ā n ) such that there are P n 1 A n 1 and p n 1 : A n X n L n 1(P n 1 ) X n making commutative a certain diagram. Consider now two objects (B n,b n ), (B n,b n ) in H n (B ) and an arrow [B n 1, [A n,a n ]] : Γ(B n,b n ) Γ(B n,b n )

298 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 in Ker Δ n. We put Y n 1 = n 1 (P n 1 ) B n 1 and we define y n 1 by the following composition: B n a n n (A n ) M n 1 (B n 1 ) B n n(p n 1 ) 1 1 n (L n 1 (P n 1 )) M n 1 (B n 1 ) B n λ n 1(P n 1 ) 1 1 M n 1 ( n 1 (P n 1 )) M n 1 (B n 1 ) B n M n 1 ( n 1 (P n 1 ) B n 1 ) B n. Then [Y n 1,y n 1 ] : (B n,b n ) (B n,b n ) is an arrow in H n (B ). inally, to check that Γ[Y n 1,y n 1 ]=[B n 1, [A n,a n ]],weputb n 2 = I, A n 1 = P n 1,a n 1 = 1 and ā n 1 = p n 1. Construction of Ψ n : given an object (B n B n, [A n+1 A n+1,a n+1 : M n (B n ) n+1 (A n+1 )]) in H n (C ), if we apply Δ n and H n+1 ( ) we obtain the following object of H n+1 (B ): ( n+1 (A n+1 ) A n+2, λ 1 n+1 (A n+1) n+2 (t n+2 ) : M n+1 ( n+1 (A n+1 )) n+2 (L n+1 (A n+1 )) n+2 (I) I). Such an object is naturally isomorphic to (I B n+1,m n+1 (I) I), which is the unit object in H n+1 (B ), via the morphism Ψ n (B n, [A n+1,a n+1 ]) =[B n B n,a 1 n+1 : n+1(a n+1 ) M n (B n )]. Remark 4.3. At this point, the reader probably wonders why we define the cohomology categorical groups of a complex using the relative kernels and relative cokernels, instead of the usual kernels and cokernels. The reason is the construction of the functor Δ n involved in the previous theorem: such a functor does not exist if we define cohomology using the usual kernel and cokernel. To make clear the problem, imagine to define H n (C ) using the usual kernel and cokernel, so that an object in H n (C ) is just an object of Ker N n.now, given an object (C n C n,c n : N n (C n ) I) in Ker N n, we look for an object Δ n (C n,c n ) = (A n+1,a n+1 : L n+1 (A n+1 ))

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 299 in Ker L n+1. Since n : B n C n is essentially surjective, there are an object B n B n and an arrow b n : n (B n ) C n, so that (M n (B n ), μ n (B n ) N n (b n ) c n : n+1 (M n (B n )) I) is an object in Ker n+1. Since ( n+1, n+1, n+1 ) is 2-exact, there are an object A n+1 A n+1 and an arrow x n+1 : ( n+1 (A n+1 ), n+1 (A n+1 )) (M n (B n ), μ n (B n ) N n (b n ) c n ) in Ker n+1. It remains to find an arrow a n+1 : L n+1 (A n+1 ) I in A n+2. Since ( n+2, n+2, n+2 ) is 2-exact, it is enough to find an arrow τ : ( n+2 (L n+1 (A n+1 )), n+2 (L n+1 (A n+1 ))) ( n+2 (I), n+2 (I)) in Ker n+2. We could take as τ the following composition λ n+1 (A n+1 ) M n+1 (x n+1 ) β n (B n ) : n+2 (L n+1 (A n+1 )) M n+1 ( n+1 (A n+1 )) M n+1 (M n (B n )) I n+2 (I). Now, to check that τ is an arrow in Ker n+2 amounts to check the commutativity of the following diagram N n+1 (c n ) N n+1 (N n (C n )) N n+1 (I ) γ n (C n ) I which precisely means that (C n,c n ) is indeed an object of the relative kernel Ker(N n, γ n ). 5. Examples and applications 5.1. Complexes of abelian groups irst of all, let us point out that, when the complex of symmetric categorical groups is in fact a complex of abelian groups, then we get the usual cohomology groups applying π and π 1 to the cohomology categorical groups. More precisely, consider a complex of abelian groups l l 1 l 2 A = A A 1 A 2 ln 1 l n+1 A n A n+1 with cohomology groups H n (A ) = Ker(l n )/Im(l n 1 ). We can construct two complexes of symmetric categorical groups: A []=A [] l [] A 1 [] l 1[] A 2 []... A [1]=A [1] l [1] A 1 [1] l 1[1] A 2 [1]... l n

3 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 Proposition 5.1. With the previous notations, we have 1. π (H n (A [])) = H n (A ) = π 1 (H n+1 (A [])), 2. π (H n (A [1])) = H n+1 (A ) = π 1 (H n+1 (A [1])). Proof. We check only part 1 because the proof of part 2 is similar. If we specialize the description of H n (A ) given in Section 3 to the case of A =A [], we have that the objects are the elements of Ker(l n ), and a pre-morphism a n a n is an element x n 1 A n 1 such that a n = l n 1 (x n 1 ) + a n. It is now clear that π (H n (A [])) = H n (A ). 5.2. Takeuchi Ulbrich cohomology Consider a complex of symmetric categorical groups L L = 1 1 2... α Each object X n 1 A n 1 gives rise to an object (L n 1 (X n 1 ), α n 1 (X n 1 )) Ker(L n, α n ). The isomorphism classes of these objects constitute a subgroup of the group of connected components π (Ker(L n, α n )). rom [15,16], we recall the following definition. Definition 5.2. With the previous notations, the nth Takeuchi Ulbrich cohomology group of the complex A is the quotient group H n U (A ) = π (Ker(L n, α n ))/ [L n 1 (X n 1 ), α n 1 (X n 1 )] Xn 1 A n 1. Proposition 5.3. With the previous notations, we have group isomorphisms π (H n (A )) H n U (A ) π 1 (H n+1 (A )). Proof. Explicitly, π (Ker(L n, α n ))/ [L n 1 (X n 1 ), α n 1 (X n 1 )] Xn 1 A n 1 is the group of equivalence classes of pairs (A n A n,a n : L n (A n ) I)such that L n+1 (a n )=α n (A n ). Two pairs (A n,a n ) and (A n,a n ) are equivalent if there is X n 1 A n 1 such that (A n,a n ) and (L n 1 (X n 1 ), α n 1 (X n 1 )) (A n,a n ) are isomorphicin Ker(L n, α n ). This amounts to ask that there is x n 1 : A n L n 1 (X n 1 ) A n making commutative the following diagram L n (A n ) a n I I I L n (x n 1 ) L n (L n 1 (X n 1 ) A n ) α n 1 (X n 1 ) a n L n (L n 1 (X n 1 )) L n (A n ) If we look now at the description of H n (A ) given in Section 3, it is clear that the previous description corresponds to π (H n (A )).

Since the functor A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 31 π : SC Abelian roups sends 2-exact sequences into exact sequences (and π 1 also, see [19]), from Theorem 4.2 and Proposition 5.3 we get the following corollary. Corollary 5.4. Let be an extension of complexes of symmetric categorical groups. There is a long exact sequence of abelian groups HU n (A ) HU n (B ) HU n (C ) HU n+1 (A ) 5.3. Ulbrich exact sequence If B is a symmetric categorical group, we can construct a canonical extension π 1 ()[1] π ()[] = where π 1 (B)[1] B is just the inclusion, and B π (B)[] sends an object on its isomorphism class (see [1]). Starting from a complex B of symmetric categorical groups and repeating the previous construction at each degree, we obtain an extension of complexes π 1 (B )[1] B π (B )[] and we can apply Theorem 4.2. Using Proposition 5.1 and Proposition 5.3 to calculate π of the 2-exact sequence of cohomology categorical groups, we get the following corollary, which is the main general result contained in [17]. Corollary 5.5. Let B be a complex of symmetric categorical groups. There is a long exact sequence of abelian groups...h n+1 (π 1 (B )) H n U (B ) H n (π (B )) H n+2 (π 1 (B ))... 5.4. Hattori Villamayor Zelinsky exact sequence If C is any (symmetric) monoidal category, the Picard categorical group Pic(C) is the (symmetric) categorical group of invertible objects and isomorphisms in C. In particular, if R is a commutative ring with unit, Pic(R) is by definition Pic(R-mod). It follows that

32 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 π (Pic(R)) is the usual Picard group of R, and π 1 (Pic(R)) is the group of units of R. Moreover, each ring homomorphism f : R S induces a monoidal functor R-mod S-mod and then a morphism of symmetric categorical groups (denoted with the same name) f : Pic(R) Pic(S). Starting from the ring homomorphism f : R S, we can construct the nth tensor power S n = S R S R R S. Moreover, for each n,wehaven + 1 face homomorphisms f i : S n S n+1 determined by f i (s 1 s n )=s 1 s i 1 s i+1 s n. The induced morphisms of symmetric categorical groups f i : Pic(S n ) Pic(S n+1 ) can be pasted together to obtain a complex Pic(S ): L n 1...Pic(S n 1 ) Pic(S n ) L n Pic(S n+1 )... where L n is a kind of alternating tensor product: L n (X) = f 1 (X) f 2 (X) f 3 (X). If we apply Corollary 5.5 to the complex Pic(S ), we obtain the Hattori Villamayor Zelinsky sequence [9,18], that is the U-Pic-exact sequence associated with the ring homomorphism f : R S (notations of Theorem 4.14 in [18], but ours H n (S/R, U) and H n (S/R, P ic) are their H n 1 )...H n+1 (S/R, U) H n U (S/R) H n (S/R, P ic) H n+2 (S/R, U)... (see also Theorem 6.1.3 in [2]). 5.5. Takeuchi exact sequence If C is a symmetric monoidal category with stable coequalizers, a new symmetric monoidal category Bim(C) can be obtained by taking as objects C-monoids and as arrows isomorphism classes of bimodules. The Brauer categorical group of C is by definition Br(C) = Pic(Bim(C)) (see [19]). If R is a commutative ring with unit, we put Br(R)= Br(R-mod). One has that π (Br(R)) is the usual Brauer group of R and π 1 (Br(R)) is the Picard group of R. Once again, a ring homomorphism f : R S induces a morphism of symmetric categorical groups Br(R) Br(S). Working in the same way as in the previous subsection, we get a complex of symmetric categorical groups Br(S ): L n 1...Br(S n 1 ) Br(S n ) L n Br(S n+1 )....

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 33 If we apply Corollary 5.5 to the complex Br(S ), we obtain the Takeuchi sequence [13], that is the Picard Brauer exact sequence associated with the ring homomorphism f : R S (notations of Theorem 6.4.2 in [2])...H n+1 (S/R, P ic) H n U (S/R, A = ) H n (S/R, Br) H n+2 (S/R,Pic).... 5.6. Simplicial cohomology, I iven a simplicial set X with degeneracies δ i : X n+1 X n, i =,...,n+ 1 and a symmetric categorical group A, following [11,4] we can construct a cosimplicial complex A X of symmetric categorical groups and strict homomorphisms: A X n is the symmetric categorical group of functors from the discrete groupoid X n to A, under pointwise tensor product; the codegeneracies are given by composition with the degeneracies d i = δ i : A X n A X n+1 ; i =,...,n+ 1. Now, by taking alternating tensor product we get a complex of symmetric categorical groups C(A X ):...A X n 1 L n 1 A X n L n A X n+1... with L n (H ) = d (H ) d 1 (H ) d 2 (H ).... The cohomology categorical groups of this complex are denoted by H n (X, A). Since a discrete groupoid X is projective with respect to essentially surjective functors, any extension in SC gives rise to a new extension X.. X. X By Theorem 4.2, we get the following corollary.

34 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 Corollary 5.6. Let be an extension of symmetric categorical groups, and fix a simplicial set X. There is a long 2-exact sequence of symmetric categorical groups... H n (X, A) H n (X, B) H n (X, C) H n+1 (X, A)... Applying the functor π : SC Abelian roups to the previous 2-exact sequence, we get the long exact sequence of abelian groups obtained in [4], Proposition 2.4. 5.7. Simplicial cohomology, II Let D be a category. As simplicial set X, we can take the nerve Ner(D) of D. Proposition 5.7. Let D be a category and A a symmetric categorical group. There is an equivalence of symmetric categorical groups Hom Cat (D, A) H (Ner(D), A). Proof. Indeed, an object of H (Ner(D), A) is a pair (A,a ), where A is a map from the objects of D to those of A, and a associates to any arrow f : X Y in D an arrow a (f ) : A (X) A (Y ) I. To such an arrow canonically corresponds an arrow ã (f ) : A (X) A (Y ), and the condition L 1 (a ) = α (A ) gives that the pair (A, ã ) is a functor from D to A. (In fact, the condition L 1 (a ) = α (A ) means that ã preserves the composition. This implies that it preserves also the identity arrows, because A is a groupoid.) If D is a category and A a categorical group, the groupoid Tors(D, A) of D-torsors under A has been studied in [5].AD-torsor under A is a rothendieck cofibration p : E D such that, for any X D, the fibre category E X is equivalent to A via a given action of A on E. The arrows in Tors(D, A) are the A-equivariant D-functors. This groupoid is a 2-groupoid adding as 2-cells the A-equivariant D-homotopies. If A is symmetric, the next proposition provides the classifying groupoid of Tors(D, A) with a structure of symmetric categorical group. Proposition 5.8. Let D be a category and A a symmetric categorical group. There is an equivalence of groupoids cl(t ors(d, A)) H 1 (Ner(D), A).

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 35 Proof. We limit our proof to the construction of the morphism H 1 (Ner(D), A) cl(t ors(d, A)). The objects to H 1 (Ner(D), A) are the systems (A f,t g,f ) consisting of for any morphism f : X Y of D, an object A f A, and for any pair of composable morphisms X f Y g Z in D, a morphism t g,f : A f A g A fg in A, which satisfy a cocycle condition. So, an object of H 1 (Ner(D), A) can be identified with a 2-cocycle in D with coefficients in A (see [5]). Thus any object (A, t) of H 1 (Ner(D), A) defines a pseudo-functor and, following the rothendieck construction, has canonically associated a cofibration P : E (A,t) D. In Theorem 4.9 in [5] it is proved that E (A,t) is in fact a D-torsor under A. A pre-arrow : (A, t) (A,t ) in H 1 (Ner(D), A) is a system = ( X, f ) consisting of for any object X D, an object X A, and for any morphism f : X Y in D, a morphism f : A f Y X A f in A which makes certain diagrams commutative. A pre-arrow : (A, t) (A,t ) defines an A-equivariant D-functor E : E (A,t) E (A,t ) which sends an object (B A,X D) E (A,t) to (B X,X). Two pre-arrows, : (A, t) (A,t ) of H 1 (Ner(D), A) are identified if there is a collection of morphisms ν ={ν X : X X X D} making a certain diagram commutative. It is easy to get an homotopy E ν : E E from such a collection ν. Corollary 5.9. Let be an extension of symmetric categorical groups, and fix a category D. There is a 2-exact sequence of symmetric categorical groups Hom Cat (,) Hom Cat (,) Hom Cat (,) cl(tors(,)) cl(tors(,)) cl(tors(,)) If D=D[1] for D a group and A=A[1] fora an abelian group, then π (cl(t ors(d, A)))= Ext cen (D, A), the group of equivalence classes of central extensions of D by A

36 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 (Example 3.9 in [5]). So, applying the functor π to the previous 2-exact sequence, we get an exact sequence of abelian groups involving the groups of central extensions. 5.8. Simplicial cohomology, III In [3], the nerve Ner 2 (D) of a categorical group D has been introduced. Let us recall that Ner 2 (D) is the 3-coskeleton of the following truncated simplicial set: Ner 2 (D) ={}, Ner 2 (D) 1 = Obj(D), Ner 2 (D) 2 ={(x, D,D 1,D 2 ) Mor(D) Obj(D) 3 x : D D 2 D 1 }, Ner 2 (D) 3 is the set of commutative diagrams in D of the form 1 x D D 3 D 3 23 D D 13 x 1 x 1 D 2 D 23 x 2 D 11 Proposition 5.1. Let D be a categorical group and A a symmetric categorical group. There is an equivalence of symmetric categorical groups Hom C (D, A) H 1 (Ner 2 (D), A). Proof. Let us restrict ourselves to the description of objects.an object of H 1 (Ner 2 (D), A) is a system (A D,a x ) consisting of for any object D D, an object A D A, and for any morphism x : D D 2 D 1 in D, a morphism a x : A D A D2 A D1 in A such that, for all (x,x 1,x 2,x 3 ) Ner 2 (D) 3, the following diagram commutes A D A D3 A D23 a x 1 A D2 A D23 1 a x3 a x2 A D A D13 a x 1 A D11 Thus, we have a monoidal functor A : D A defined by A(D) = A D, with canonical morphisms given by a 1D D 2 : A D A D2 A D D 2. inally, if the categorical group D is symmetric, it is possible to refine again its nerve to take into account the symmetric structure. We refer to [3] for a detailed description of the nerve Ner 3 (D) of a symmetric categorical group D.

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 37 Proposition 5.11. Let D and A be symmetric categorical groups. There is an equivalence of symmetric categorical groups Hom SC (D, A) H 2 (Ner 3 (D), A). 6. The kernel-cokernel lemma In this section, we obtain the kernel-cokernel (or snake ) lemma for symmetric categorical groups as a particular case of the long cohomology sequence of Theorem 4.2. We will then apply the lemma to get a low-dimensional cohomology sequence involving derivations of categorical groups. 6.1. The kernel-cokernel lemma for symmetric categorical groups We start with two general lemmas on symmetric categorical groups. Lemma 6.1. Consider the following diagram in SC Ker L e ψ ε M E N Ker e ε where L and ψ are induced by the universal property of Ker (so that ψ,, ε and ε are compatible). 1. If N is full and faithful, then the left-hand square is a bi-pullback; 2. If, moreover, M is full (faithful)(essentially surjective), then L is full (faithful)(essentially surjective). Proof. 1. rom [1], Proposition 5.2, recall that N is full and faithful iff for all SC, the functor Hom SC (,N)is full and faithful. Using this fact, the proof is a (long) argument on bi-limits which holds in any 2-category. 2. It follows from the first part, using the stability under bi-pullback of the involved classes of morphisms (see Proposition 5.2 in [1]).

38 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 Lemma 6.2. Consider the following diagram in SC L ψ M π P Coker N π P Coker where N and are induced by the universal property of Coker (so that ψ,, π and π are compatible). 1. If L is full and essentially surjective, then the right-hand square is a bipushout; 2. If, moreover, M is full (faithful) (essentially surjective), then N is full (faithful) (essentially surjective). Proof. Dual of the previous one: by Proposition 5.3 in [1], L is full and essentially surjective iff for all SC, the functor Hom SC (N, ) is full and faithful; the stability under bi-pushout is established in [1], Proposition 5.1. ix now the following diagram in SC L λ M μ N ' ' ' ' ' ' (5) where (,,) and (,, ) are 2-exact sequences, is essentially surjective and is faithful. We assume also that,, λ and μ are compatible (as at the beginning of Section 4). Proposition 6.3 (The kernel-cokernel lemma). There are a morphism and two 2-cells in SC Δ : Ker N Coker L Σ : Ḡ Δ Ψ : Δ

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 39 making the following sequence 2-exact in each point Ker L Ker M Coker L Σ Δ ψ Ker N ' ' Coker M ' Coker N Proof. Consider the factorization of as a full and essentially surjective functor 1 followed by a faithful functor 2 (Proposition 2.1 in [1]). Consider also the factorization of as an essentially surjective functor 1 followed by a full and faithful functor 2 (Proposition 2.3 in [1]) L 1 α H 2 α M β K β 1 2 N Since 1 is orthogonal to (Proposition 4.3 in [1]) and is orthogonal to 2 (Proposition 4.6 in [1]), we get the fill-in H,α, α and K,β, β as in the previous diagram. Moreover, since 1 is full and essentially surjective, there is a unique 2-cell ψ : 2 such that 1 ψ = ; since 2 is full and faithful, there is a unique 2-cell ψ : 1 such that ψ 2 =. In this way, we have constructed a new diagram in SC H 2 ψ ψ α M (β ) 1 1 K (6) Composing with 1 and 2, we can check the compatibility of the 2-cells in (6) using that of the 2-cells in (5). Moreover, ( 2, ψ,)is 2-exact (and then it is an extension) because, by Lemma 6.2, the cokernel of 2 is equivalent to the cokernel of. Analogously, (, ψ, 1 ) is 2-exact because, by Lemma 6.1, the kernel of 1 is equivalent to the kernel of. Now, adding zero-morphisms and canonical 2-cells, we can turn the morphism of extensions (6) into an extension of complexes. The only non trivial cohomology categorical groups of these complexes are the (usual) kernels and cokernels of H, M and K. Therefore,

31 A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 Theorem 4.2 gives us the following 2-exact sequence Ker H Ker M Ker K Coker H Coker M Coker K Observe now that, by Lemma 6.1, Ker K and Ker N are equivalent, and, by Lemma 6.2, Coker H and Coker L are equivalent. Moreover, by Lemma 6.1 again, the comparison Ker L Ker H is full and essentially surjective, so that the 2-exactness of Ker H Ker M Ker K implies the 2-exactness of Ker L Ker M Ker K. In the same way, by Lemma 6.2 the comparison Coker K Coker N is full and faithful, so that Coker H Coker M Coker N is 2-exact. inally, we have proved the 2-exactness of Ker L Ker M Ker N Coker L Coker M Coker N 6.2. Derivations of categorical groups To end, we explain how the low-dimensional cohomology sequence obtained in [7], Theorem 6.2, is a special case of the 2-exact sequence of Proposition 6.3. or detailed definitions about derivations of categorical groups, we refer to [7,8]. ix a categorical group and a symmetric -module B with action : B B. A derivation is a functor D : B together with a natural and coherent family of isomorphisms δ X,Y : D(X) X D(Y) D(X Y). Derivations and their morphisms give rise to a groupoid Der(, B), which is a symmetric categorical group under pointwise tensor product. (Observe that, in general, if the -module B is only braided, the categorical group Der(, B) is no longer braided.) This construction plainly extends to a 2-functor from the 2-category of symmetric -modules and equivariant morphisms to SC. Moreover, for any symmetric -modules and equivariant morphisms to SC. Moreover, for any symmetric -module B, there is an inner derivation morphism I : B Der(, B) I(B) : B I(B)(X) = X B B whose kernel and cokernel are denoted by H (, B) and H 1 (, B) and called the lowdimensional cohomology categorical groups of with coefficients in B. Now, if : A B is an equivariant morphism of symmetric -modules, its equivariant structure induces

A. del Río et al. / Journal of Pure and Applied Algebra 196 (25) 279 312 311 a 2-cell in SC A I Der(, A) inally, if λ B I Der(, B) is an extension of symmetric -modules, by Proposition 3.4 in [8] we get a diagram in SC I λ I Der(, ) Der(, ) Der(, )... with (,, ) 2-exact and faithful. Since it is straightforward to check the compatibility of λ, μ, and, as a corollary of Proposition 6.3 we get the 2-exact cohomology sequence μ H (, A) H (, B) H (, C) H 1 (, A) H 1 (, B) H 1 (, C). (7) If is a discrete categorical group, and A, B and C are discrete -modules, then applying π to the previous sequence we recover the familiar exact sequence of low-dimensional cohomology groups. Several other particular cases of interest are discussed in [7]. The non symmetric analogue of the 2-exact sequence (7) is studied in [6]. I References [1] D. Bourn, E.M. Vitale, Extensions of symmetriccat-groups, Homol. Homotopy Appl. 4 (22) 13 162 (http://www.rmi.acnet.ge/hha/). [2] S. Caenepeel, Brauer groups, Hopf Algebras and alois Theory, Kluwer Academic Publishers, Dordrecht, 1998. [3] P. Carrasco, A.M. Cegarra, (Braided) Tensor structures on homotopy groupoids and nerves of (braided) categorical groups, Commun. Algebra 24 (1996) 3995 458.

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