A comonadic interpretation of Baues-Ellis homology of crossed modules
aa r X i v : . [ m a t h . K T ] M a y A COMONADIC INTERPRETATION OFBAUES–ELLIS HOMOLOGY OF CROSSED MODULES
GURAM DONADZE AND TIM VAN DER LINDEN
Abstract.
We introduce and study a homology theory of crossed moduleswith coefficients in an abelian crossed module. We discuss the basic propertiesof these new homology groups and give some applications. We then restrictour attention to the case of integral coefficients. In this case we regain thehomology of crossed modules originally defined by Baues and further developedby Ellis. We show that it is an instance of Barr–Beck comonadic homology, sothat we may use a result of Everaert and Gran to obtain Hopf formulae in alldimensions. Introduction
The concept of a crossed module originates in the work of Whitehead [35]. It wasintroduced as an algebraic model for path-connected CW spaces whose homotopygroups are trivial in dimensions strictly above . The role of crossed modules andtheir importance in algebraic topology and homological algebra are nicely demon-strated in the articles [6, 8, 9, 27, 30, 35], just to mention a few. The study of crossedmodules as algebraic objects in their own right has been the subject of many invest-igations, amongst which the Ph.D. thesis of Norrie [32] is prominent. She showedthat some group-theoretical concepts and results have suitable counterparts in thecategory of crossed modules.In [2], Baues defined a (co)homology theory of crossed modules via classifyingspaces, which was studied using algebraic methods by Ellis in [14]. A differentapproach to (co)homology of crossed modules is due to Carrasco, Cegarra andR.-Grandjeán. In particular, in the paper [10], the authors develop Barr–Beck co-monadic homology [1] of crossed modules, relative to the canonical comonad whicharises from the forgetful adjunction to the category of sets, and with coefficients inthe abelianisation functor. In the article [23] certain relationships between thesetwo homology theories are explored.In the present article we introduce and study a new homology theory of crossedmodules—one where coefficients are taken in an abelian crossed module. In the caseof integral coefficients, it restricts to the theory of Baues and Ellis. We furthermoreshow that, in this particular case, homology forms a non-abelian derived functor,which may be obtained via comonadic homology as in the approach of Carrasco, Date : 22nd May 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Comonadic homology; crossed module; spectral sequence; Hopfformula.This work was partially supported by Agencia Estatal de Investigación (Spain), grantMTM2016-79661-P (EU supported included).The first author wishes to thank Prof. Manuel Ladra González and the University of Santiagode Compostela for their kind hospitality.The second author is a Research Associate of the Fonds de la Recherche Scientifique–FNRS. Hewishes to thank Dr Viji J. Thomas with the Indian Institute of Science Education and Researchat Thiruvananthapuram, and Prof. Manuel Ladra González with the University of Santiago deCompostela for their kind hospitality.
Cegarra and R.-Grandjeán. Since the functor which is being derived turns out tobe “sufficiently nice”, we may apply results of Everaert and Gran [17] leading to aninterpretation of the homology objects via higher Hopf formulae, extending thoseof Brown and Ellis [7, 13, 18, 15].Let us give a quick overview of the results in the paper. Consider a crossed mod-ule p H, G, µ : H Ñ G q acting on an abelian crossed module p C, A, ν q ; this meansthat p C, A, ν q carries the structure of a Beck module [3] over p H, G, µ q . Then wedenote by H n pp H, G, µ q , p C, A, ν qq the n th homology group of p H, G, µ q with coef-ficients in p C, A, ν q , defined as the homology of the total complex of some suitabledouble complex—see Section 3 for details. If p C, A, ν q “ p , Z , q and p H, G, µ q actstrivially on p , Z , q , we regain the homology of p H, G, µ q with integral coefficientsas defined by Baues and Ellis.For an inclusion crossed module p N, G, σ q acting on an abelian crossed module p C, A, ν q , we obtain the exact sequence of abelian groups ¨ ¨ ¨ Ñ H n ´ p G { N, Ker ν q Ñ H n pp N, G, σ q , p C, A, ν qq Ñ H n p G { N, A { Im ν qÑ H n ´ p G { N, Ker ν q Ñ ¨ ¨ ¨ where H n p G { N, Ker ν q and H n p G { N, A { Im ν q denote the Eilenberg–Mac Lane ho-mologies of G { N with coefficients in Ker ν and in A { Im ν , respectively: see Pro-position 3.10.A first main result of this paper is the Lyndon–Hochschild–Serre spectral se-quence adapted to crossed modules (Theorem 4.3). We use it to show that if twocrossed modules are (weakly) homotopically equivalent, then their homology groupsare isomorphic (Proposition 4.5). Another consequence of this spectral sequenceis a five-term exact sequence relating the homology groups of crossed modules inlow degrees (Proposition 4.6). We also prove that the third homology of a crossedmodule (as defined by Baues and Ellis) is isomorphic to the first homology of thesame crossed module with certain coefficients (Proposition 4.9).In Section 5 we show that the homology groups (with integral coefficients) ofa crossed module can be described via non-abelian left derived functors (The-orem 5.2). The comonad on the category of crossed modules is still the one inducedby the adjunction to sets as in [10]. However, the functor which is being derivedis different: it sends a crossed module p H, G, µ q to the group G {p µ p H q ¨ r G, G sq ,where µ p H q ⊳ G is the image of µ and r G, G s ⊳ G is the ordinary commutator sub-group. This functor A : XMod Ñ Ab turns out to be an example considered in [17].The theory developed there leads to an interpretation of Baues–Ellis homology viahigher Hopf formulae (Theorem 6.1).2. Crossed modules, their nerves and their actions A crossed module p H, G, µ q is a homomorphism of groups µ : H Ñ G , togetherwith an action of G on H , such that the identities µ p g h q “ gµ p h q g ´ (Precrossed module identity) µ p h q h “ hh h ´ (Peiffer identity)hold for all g P G and h , h P H . A common instance of a crossed module is thatof a group G possessing a normal subgroup N ⊳ G ; the inclusion homomorphism N ã Ñ G is a crossed module with G acting on N by the conjugation in G , calledthe inclusion crossed module of N ⊳ G . More generally, given any crossedmodule p H, G, µ q , we may consider the semidirect product H ¸ G . Then there arehomomorphisms s : H ¸ G Ñ G : p h, g q ÞÑ g , t : H ¸ G Ñ G : p h, g q ÞÑ µ p h q g and e : G Ñ H ¸ G : g Ñ p , g q and a binary operation p h , g q ˝ p h, g q “ p hh , g q for allpairs p h, g q , p h , g q P H ¸ G such that µ p h q g “ g . This composition ˝ with the COMONADIC INTERPRETATION OF BAUES–ELLIS HOMOLOGY 3 source map s , target map t and identity map e constitutes an internal categoryin the category of groups. Conversely, given any internal category p M, G, s, t, e q in Gp , the induced map tk : Ker s Ñ G where k : Ker s Ñ M denotes the kernelof s , together with the restriction to G of the conjugation action of M on Ker s ,forms a crossed module.A morphism of crossed modules p ρ, ν q : p H, G, µ q Ñ p H , G , µ q is a com-mutative square of groups H µ , ρ (cid:12) (cid:18) G ν (cid:12) (cid:18) H µ , G such that ρ p g h q “ ν p g q ρ p h q for all g P G , h P H . The category of crossed moduleswith their morphisms is denoted XMod . It is well known to be a variety of Ω -groupsin the sense of Higgins [24, 10] and as such it is an example of a semi-abelian varietyof algebras [26]. Furthermore [27, 29], it is equivalent to the category Cat p Gp q ofinternal categories in the category of groups, essentially via the explanation recalledabove.Given a crossed module p H, G, µ q , the nerve of its category structure forms thesimplicial group Ner ˚ p H, G, µ q , called the nerve of the crossed module p H, G, µ q .It is not difficult to see that Ner n p H, G, µ q “ H ¸p¨ ¨ ¨ p H ¸ G q ¨ ¨ ¨ q with n semidirectfactors of H , and that the face and degeneracy homomorphisms are given by d p h , . . . , h n , g q “ p h , . . . , h n , g q ,d i p h , . . . , h n , g q “ p h , . . . , h i h i ` , . . . , h n , g q , ă i ă n,d n p h , . . . , h n , g q “ p h , . . . , h n ´ , µ p h n q g q ,s i p h , . . . , h n , g q “ p h , . . . , h i , , h i ` , . . . , h n , g q , ď i ď n. Given a simplicial group G ˚ “ p G n , d in , s in q , recall that its Moore normalisa-tion is a chain complex of groups N G ˚ “ p N G n , B n q , where N G ˚ “ G while N n G ˚ “ n ´ č i “ Ker d ni and B n “ d nn | N n G ˚ , n ě . For n ě , the n th homotopy group of G ˚ is defined as π n p G ˚ q “ Ker B n { Im B n ` . When an augmented simplicial group p G ˚ , d , G q is given, then in dimension zerowe calculate the homotopy group as π p G ˚ , d , G q “ Ker d { Im B . We say thatthe augmented simplicial group p G ˚ , d , G q is aspherical if π n p G ˚ , d , G q “ forall n ě and Im d “ G .Given a crossed module p H, G, µ q , the Moore complex of its nerve is trivial indimensions ě . In fact the Moore complex of Ner ˚ p H, G, µ q is just the originalcrossed module up to isomorphism with H in dimension and G in dimension .Consequently, we have the isomorphisms π p Ner ˚ p H, G, µ qq –
Ker µ and π p Ner ˚ p H, G, µ qq – G { Im µ. ( A )Suppose we are given composable morphisms p ρ, ν q : p H, G, µ q Ñ p H , G , µ q and p ρ , ν q : p H , G , µ q Ñ p H , G , µ q such that Ker ρ “ ρ and Ker ν “ ν , while Coker ρ “ ρ and Coker ν “ ν in Gp . This is precisely saying that Ker p ρ , ν q “p ρ, ν q and Coker p ρ, ν q “ p ρ , ν q in XMod , so that p ρ, ν q and p ρ , ν q form a short GURAM DONADZE AND TIM VAN DER LINDEN exact sequence of crossed modules. In this case we write , p H, G, µ q p ρ,ν q , p H , G , µ q p ρ ,ν q , p H , G , µ q , . Recall that the nerve functor is exact, sending any short exact sequence of crossedmodules to a short exact sequence of simplicial groups.If both G and H are abelian groups with trivial action of G on H , then p H, G, µ q is said to be an abelian crossed module. A crossed module is abelian precisely whenit is an an abelian group object in the category of crossed modules [10]. Given acrossed module p H, G, µ q , a Beck module [3] over p H, G, µ q is an abelian groupobject in the slice category p XMod
Ó p
H, G, µ qq . Since the category of crossedmodules satisfies the so-called Smith is Huq condition [31, 12], such an internalabelian group object is the same thing as a split exact sequence , p M, P, ν q , p H , G , µ q , p H, G, µ q l r , . where p M, P, ν q is an abelian crossed module. As always in a semi-abelian category,the middle object p H , G , µ q in such a split exact sequence may be seen as asemidirect product p M, P, ν q ¸ ξ p H, G, µ q where ξ is an internal action [5, 4] of p H, G, µ q on p M, P, ν q . In the case of crossed modules, these internal actions agreewith the concept of an action studied by Norrie [33] and Forrester-Barker [20]. Herewe recall the presentation given in [11]. An action of a crossed module p H, G, µ q onanother crossed module p M, P, ν q is completely determined by an action of G (andso H ) on M and P together with a map ξ : H ˆ P Ñ M such that the conditions ν p g m q “ g ν p m q , g p p m q “ g p m,νξ p h, p q “ µ p h q pp ´ , ξ p h, ν p m qq “ µ p h q mm ´ , g ξ p h, p q “ ξ p g h, g p q , ξ p hh , p q “ µ p h q ξ p h , p q ξ p h, p q ,ξ p h, pp q “ ξ p h, p q p ξ p h, p q hold for all g P G , h , h P H , p , p P P , m P M . For instance, the trivial actionof G on M and P together with the trivial map ξ : H ˆ P Ñ M provides an actionof p H, G, µ q on p M, P, ν q , called the trivial action .Thus a morphism of p H, G, µ q -modules p α, β q : p M, P, ν q Ñ p M , P , ν q amountsto a pair of G -module homomorphisms α : M Ñ M , β : P Ñ P such that βν “ ν α and αξ “ ξ p H ˆ β q . Note that given an action of p H, G, µ q on p M, P, ν q and amorphism of crossed modules p ¯ H, ¯ G, ¯ µ q Ñ p H, G, µ q , we naturally obtain an actionof p ¯ H, ¯ G, ¯ µ q on p M, P, ν q . Moreover, the conditions mentioned above imply that theaction of H on Ker ν is trivial, while the action of H on P is trivial modulo Im ν .Thus, we have naturally defined actions of G { Im µ on both Ker ν and P { Im ν .3. Homology of crossed modules with coefficients
Any action of a crossed module p H, G, µ q on another crossed module p M, P, ν q induces a (pointwise) action of the simplicial group Ner ˚ p H, G, µ q on Ner ˚ p M, P, ν q .Indeed, actions correspond to split extensions, and since the nerve functor preservesall limits, it does also preserve split extensions, and the corresponding actions.Given a group G , let B ˚ p G q “ p B ˚ p G q , Bq denote the standard G -resolutionover Z : we set B ´ p G q “ Z , and for m ě , B m p G q “ Z p G q b Z p G q b ¨ ¨ ¨ b Z p G q loooooooooooooooomoooooooooooooooon m ` factors COMONADIC INTERPRETATION OF BAUES–ELLIS HOMOLOGY 5 where Z p G q denotes the group ring over G , and Bp g b g b ¨ ¨ ¨ b g m q is p´ q m g b g b ¨ ¨ ¨ b g m ´ ` m ´ ÿ i “ p´ q i g b ¨ ¨ ¨ b g i g i ` b ¨ ¨ ¨ b g m . We know that B m p G q has a structure of a right G -module defined by p g b g b ¨ ¨ ¨ b g m q g “ p g ´ g b g b ¨ ¨ ¨ b g m q . Suppose that G ˚ “ p G n , d in , s in q is a simplicial group acting on an abelian simpli-cial group A ˚ “ p A n , B in , t in q , i.e., each A n is an abelian group. Then for each n ě we have a map of complexes B ˚ p G n q b G n A n Ñ B ˚ p G n ´ q b G n ´ A n ´ defined bythe alternating sum of face homomorphisms: b b a ÞÑ n ÿ i “ p´ q i d in p b q b B in p a q , for all b P B m p G n q , a P A n . In this way we obtain the bicomplex of abelian groups (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) B p G q b G A (cid:12) (cid:18) B p G q b G A (cid:12) (cid:18) l r B p G q b G A (cid:12) (cid:18) l r ¨ ¨ ¨ l r B p G q b G A (cid:12) (cid:18) B p G q b G A (cid:12) (cid:18) l r B p G q b G A (cid:12) (cid:18) l r ¨ ¨ ¨ l r B p G q b G A B p G q b G A l r B p G q b G A l r ¨ ¨ ¨ l r which we write B p G ˚ , A ˚ q . Definition 3.1.
For n ě , the n th homology group of G ˚ with coefficientsin A ˚ is defined by the formula H n p G ˚ , A ˚ q “ H n ` Tot B p G ˚ , A ˚ q ˘ . This immediately gives a first quadrant spectral sequence E pq “ H q p G p , A p q ñ H p ` q p G ˚ , A ˚ q , where H q p G p , A p q is the classical Eilenberg–Mac Lane homology. Definition 3.2.
Let p H, G, µ q be a crossed module acting on an abelian crossedmodule p C, A, ν q . Then, the n th homology group of p H, G, µ q with coefficientsin p C, A, ν q is defined by the formula H n pp H, G, µ q , p C, A, ν qq “ H n ` Tot B p Ner ˚ p H, G, µ q , Ner ˚ p C, A, ν qq ˘ , n ě . Suppose that a crossed module p H, G, µ q acts trivially on the inclusion crossedmodule p , Z , q . Then we write H n p H, G, µ q instead of H n pp H, G, µ q , p , Z , qq andcall it the n th homology group of p H, G, µ q with integral coefficients . Proposition 3.3.
For each n ě , H n p H, G, µ q is isomorphic to the n th homologyof the crossed module p H, G, µ q with integral coefficients considered in [2] and [14] .Proof. For any (discrete) group, one can define its nerve by viewing it as a categorywith just one object. Let N ˚˚ p H, G, µ q be the bisimplicial set obtained by formingthe nerve of the group in each dimension of the simplicial group Ner ˚ p H, G, µ q .Denote by Z ` N ˚˚ p H, G, µ q ˘ the bisimplicial abelian group obtained by formingthe free abelian group pointwise in all degrees, i.e., Z ` N pq p H, G, µ q ˘ is the free GURAM DONADZE AND TIM VAN DER LINDEN abelian group over N pq p H, G, µ q for each p ě and q ě . Then for each n ě ,the n th homology group of the diagonal of Z ` N ˚˚ p H, G, µ q ˘ is isomorphic to the n th homology group of p H, G, µ q with integral coefficients defined by Baues andEllis (see Proposition 2 in [14]). Moreover, by taking the alternating sums of facehomomorphisms of Z ` N ˚˚ p H, G, µ q ˘ in all vertical and horizontal directions, weregain the aforementioned bicomplex B p Ner ˚ p H, G, µ q , Ner ˚ p , Z , qq . Whence theproposition. (cid:3) Let G be a group acting on an abelian group A . Clearly then, the inclusioncrossed module p , G, q acts on p , A, q . Proposition 3.4.
There is an isomorphism H n pp , G, q , p , A, qq – H n p G, A q , n ě . Proof.
The bicomplex B p Ner ˚ p , G, q , Ner ˚ p , A, qq has the following form: (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) B p G q b G A (cid:12) (cid:18) B p G q b G A (cid:12) (cid:18) l r B p G q b G A (cid:12) (cid:18) l r ¨ ¨ ¨ l r B p G q b G A (cid:12) (cid:18) B p G q b G A (cid:12) (cid:18) l r B p G q b G A (cid:12) (cid:18) l r ¨ ¨ ¨ l r B p G q b G A B p G q b G A l r B p G q b G A l r ¨ ¨ ¨ l r Therefore
Tot B p Ner ˚ p , G, q , Ner ˚ p , A, qq is quasi-isomorphic to the first columnof B p Ner ˚ p , G, q , Ner ˚ p , A, qq . Whence the proposition. (cid:3) In what follows, let p C, A, ν q denote an abelian crossed module. We will writethe operations in A and C additively. Proposition 3.5.
Let p H, G, µ q be a crossed module acting on p C, A, ν q . Then H pp H, G, µ q , p C, A, ν qq “ A {p ν p C q ` x G, A yq , where x G, A y denotes the subgroup of A generated by all g a ´ a for g P G and a P A .Proof. We have a first quadrant spectral sequence E pq “ H q p Ner p p H, G, µ q , Ner p p C, A, ν qq ñ H p ` q pp H, G, µ q , p C, A, ν qq . In particular E “ H p G, A q “ A {x G, A y ,E “ H p H ¸ G, C ¸ A q “ p C ¸ A q{x H ¸ G, C ¸ A y , and the differential E Ñ E is induced by p c, a q ÞÑ ν p c q for all c P C and a P A .Thus E “ Coker p E Ñ E q “ A {p ν p C q ` x G, A yq . Moreover H pp H, G, µ q , p C, A, ν qq “ E . Whence the proposition. (cid:3) Proposition 3.6.
Let p H, G, µ q be a crossed module. Then H p H, G, µ q “ G {p µ p H q ¨ r G, G sq . COMONADIC INTERPRETATION OF BAUES–ELLIS HOMOLOGY 7
Proof.
We have a first quadrant spectral sequence E pq “ H q p Ner p p H, G, µ q , Z q ñ H p ` q p H, G, µ q . Thus E q “ Z for q ě and E q “ for q ě . This implies that H p H, G, µ q “ E . Since the first homology of a group with integral coefficients is isomorphic to theabelianisation of the same group, we have: E “ G {r G, G s ,E “ p H ¸ G q{r H ¸ G, H ¸ G s , and the differential E Ñ E is induced by p h, g q ÞÑ µ p h q . Thus E “ Coker p E Ñ E q “ G {p µ p H q ¨ r G, G sq , which finishes the proof. (cid:3) Proposition 3.7.
For any short exact sequence of p H, G, µ q -modules Ñ p
C, A, ν q Ñ p C , A , ν q Ñ p C , A , ν q Ñ , there is a long exact sequence of homology groups ¨ ¨ ¨ Ñ H n pp H, G, µ q , p C, A, ν qq Ñ H n pp H, G, µ q , p C , A , ν qq Ñ H n pp H, G, µ q , p C , A , ν qq Ñ H n ´ pp H, G, µ q , p C, A, ν qq Ñ ¨ ¨ ¨ . Proof.
It is easy to see that the short exact sequence given in the proposition yieldsthe following short exact sequence of bicomplexes: Ñ B p Ner ˚ p H, G, µ q , Ner ˚ p C, A, ν qq Ñ B p Ner ˚ p H, G, µ q , Ner ˚ p C , A , ν qq ÑÑ B p Ner ˚ p H, G, µ q , Ner ˚ p C , A , ν qq Ñ . The corresponding long exact homology sequence is the one claimed. (cid:3)
For any simplicial group G ˚ and any functor T from the category of groups to thecategory of abelian groups, let T p G ˚ q denote the abelian simplicial group obtainedby applying the functor T dimension-wise to G ˚ . The following lemma was offeredto us by Nick Inassaridze. Lemma 3.8.
Let p G ˚ , d , G q be an augmented aspherical simplicial group. Then p Z p G ˚ q , Z p d q , Z p G qq is also aspherical.Proof. We write G ´ “ G . Since p G ˚ , d , G q is aspherical, there is a sequence ofmaps (which need not be homomorphisms) p h n : G n Ñ G n ` q n ě´ which formsa left contraction for p G ˚ , d , G q ; see [25] for more details. Then the sequence p Z p h n q : Z p G n q Ñ Z p G n ` qq n ě´ will be a left contraction for p Z p G ˚ q , Z p d q , Z p G qq .Hence p Z p G ˚ q , Z p d q , Z p G qq is aspherical. (cid:3) Let G ˚ be a simplicial group acting on an abelian simplicial group A ˚ . For each m ě , let B m p G ˚ , A ˚ q denote the following abelian simplicial group: ¨ ¨ ¨ , , , , B m p G q b G A , , , B m p G q b G A , , B m p G q b G A . Lemma 3.9.
Let p G ˚ , d , G q be an augmented aspherical simplicial group. Thenthere is an isomorphism π n p B m p G ˚ , A ˚ qq – B m ´ p G q b π n p A ˚ q . GURAM DONADZE AND TIM VAN DER LINDEN
Proof.
For each m ě and k ě we have an isomorphism of abelian groups: B m p G k q b G k A k – B m ´ p G k q b A k . Thus, we need to calculate the n th homotopy group of the following simplicialabelian group: ¨ ¨ ¨ , , , , B m ´ p G q b A , , , B m ´ p G q b A , , B m ´ p G q b A . This latter simplicial group is the diagonal of the bisimplicial group (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) ¨ ¨ ¨ , , , , B m ´ p G q b A (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) , , , B m ´ p G q b A , , (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) B m ´ p G q b A (cid:12) (cid:18) (cid:12) (cid:18) (cid:12) (cid:18) ¨ ¨ ¨ , , , , B m ´ p G q b A (cid:12) (cid:18) (cid:12) (cid:18) , , , B m ´ p G q b A , , (cid:12) (cid:18) (cid:12) (cid:18) B m ´ p G q b A (cid:12) (cid:18) (cid:12) (cid:18) ¨ ¨ ¨ , , , , B m ´ p G q b A , , , B m ´ p G q b A , , B m ´ p G q b A . Thus, by the Eilenberg–Zilber theorem [34] and the Künneth formula [28] we get: π n p B m p G ˚ , A ˚ qq “ à i ` j “ n π i p B m ´ p G ˚ qq b π j p A ˚ q . Moreover, by Lemma 3.8 the augmented simplicial group p B m ´ p G ˚ q , B m ´ p d q , B m ´ p G qq is aspherical. Whence the lemma. (cid:3) Given a group G and a normal subgroup N ⊳ G , let σ : N Ñ G be the naturalinclusion. Suppose that p N, G, σ q acts on an abelian crossed module p C, A, ν q .Recall that both Ker ν and A { Im ν have a G { N -module structure (see Section 2).Thus, the homology groups H n p G { N, Ker ν q and H n p G { N, A { Im ν q , for n ě ,discussed in the next proposition make sense. Proposition 3.10.
There is a long exact sequence ¨ ¨ ¨ Ñ H n ´ p G { N, Ker ν q Ñ H n pp N, G, σ q , p C, A, ν qq Ñ H n p G { N, A { Im ν qÑ H n ´ p G { N, Ker ν q Ñ ¨ ¨ ¨ Proof.
We have a first quadrant spectral sequence E pq “ H q ` B p p Ner ˚ p N, G, σ q , Ner ˚ p C, A, ν qq ˘ ñ H p ` q pp N, G, σ q , p C, A, ν qq . By Lemma 3.9 and ( A ) we have: E pq “ $’&’% when q ě , B p ´ p G { N q b Ker ν when q “ , B p ´ p G { N q b p A { Im ν q when q “ .Furthermore, E p “ H p p B ˚´ p G { N q b Ker ν q “ H p p B ˚ p G { N q b G { N Ker ν q “ H p p G { N, Ker µ q ,E p “ H p p B ˚´ p G { N q b p A { Im ν qq “ H p p B ˚ p G { N q b G { N p A { Im ν qq“ H p p G { N, p A { Im ν qq . This implies the proposition. (cid:3)
COMONADIC INTERPRETATION OF BAUES–ELLIS HOMOLOGY 9
It is well known that the category of crossed modules
XMod may be consideredas a variety of Ω -groups in the sense of Higgins [24]. As explained in [10], theforgetful functor U : XMod Ñ Set to the category of sets assigns, to a crossedmodule p H, G, µ q , the cartesian product of the underlying sets of the groups H and G . Its left adjoint is constructed as follows. Given a group G , let G ` G bethe coproduct (= free product) of G with itself, with injections ι , ι : G Ñ G ` G ,and let i G : G Ñ G ` G be the the normal closure of ι , which may be obtained asthe kernel of the retraction p G q : G ` G Ñ G of ι . This determines an inclusioncrossed module p G, G ` G, i G q . The functor F : Set Ñ XMod assigns, to any set X ,the inclusion crossed module p F X , F X ` F X , i F X q , where F X is the free group over X . Corollary 3.11.
Let p R, F, η q be a crossed module, free over Set , acting on anabelian crossed module p C, A, µ q . Then H n pp R, F, η q , p C, A, ν qq “ , n ě ,H n p R, F, η q “ , n ě . Proof.
By the above, p R, F, η q is an inclusion crossed module. Moreover, F { R is afree group. Hence, the corollary results from Proposition 3.10. (cid:3) The Lyndon–Hochschild–Serre spectral sequence
Let G ˚ be a simplicial group acting on an abelian simplicial group A ˚ . For each m ě we have an abelian simplicial group ¨ ¨ ¨ , , , , H m p G , A q , , , H m p G , A q , , H m p G , A q , denoted by H m p G ˚ , A ˚ q . Remark . H m p G ˚ , A ˚ q is not to be confused with H m p G ˚ , A ˚ q . The latter is agroup, while H m p G ˚ , A ˚ q is a simplicial group.Given a short exact sequence of simplicial groups ÝÑ Γ ˚ ÝÑ Π ˚ ÝÑ G ˚ ÝÑ and an abelian simplicial group A ˚ together with an action of Π ˚ on A ˚ , we definean action of G ˚ on H m p Γ ˚ , A ˚ q for each m ě . First define an action of Π n on B m p Π n q b Γ n A n by the formula: x p b b a q “ bx ´ b xa, x P Π n , b P B m p Π n q , a P A n . This action is Γ n -invariant. Hence, we have an induced action of G n on the group B m p Π n q b Γ n A n . Since the complex B ˚ p Π n qb Γ n A n calculates the homology groupsof Γ n with coefficients in A n , we have derived an action of G ˚ on H m p Γ ˚ , A ˚ q , foreach m ě .The following is an analogue of the Lyndon–Hochschild–Serre spectral sequence. Proposition 4.2.
There is a first quadrant spectral sequence H p p G ˚ , H q p Γ ˚ , A ˚ qq ñ H p ` q p Π ˚ , A ˚ q . Proof.
Consider the following bicomplex: B (cid:12) (cid:18) ´B (cid:12) (cid:18) B (cid:12) (cid:18) X B (cid:12) (cid:18) X ´B (cid:12) (cid:18) l r X B (cid:12) (cid:18) l r ¨ ¨ ¨ l r X X l r X l r ¨ ¨ ¨ , l r where p X p ˚ , Bq “
Tot ` B p G p q b G p p B p Π p q b Γ p A p q ˘ . Here the tensor product oftwo complexes B p G p q and B p Π p q b Γ p A p is defined in the ordinary way. Then, thecomplex X p ˚ is quasi-isomorphic to B p Π p q b Π p A p (see [28]). Therefore, we havean isomorphism H n p Tot p X ˚˚ qq – H n p Π ˚ , A ˚ q , n ě . ( B )Furthermore, observe that Tot p X ˚˚ q n “ à p ` q ` l “ n B p p G l q b G l p B q p Π l q b Γ l A l q . We define the following filtration on
Tot p X ˚˚ q : F m p Tot p X ˚˚ q n q “ à p ` q ` l “ np ` l ď m B p p G l q b G l p B q p Π l q b Γ l A l q . The associated spectral sequence has the form E mq “ à p ` l “ m B p p G l q b G l H q p Γ l , A l q , and E mq is isomorphic to the m th homology group of the following bicomplex: (cid:12) (cid:18) (cid:12) (cid:18) B p G q b G H q p Γ , A q (cid:12) (cid:18) B p G q b G H q p Γ , A q (cid:12) (cid:18) l r ¨ ¨ ¨ l r B p G q b G H q p Γ , A q B p G q b G H q p Γ , A q l r ¨ ¨ ¨ l r By definition this bicomplex calculates the homology groups of G ˚ with coefficientsin H q p Γ ˚ , A ˚ q . Thus, E mq “ H m p G ˚ , H q p Γ ˚ , A ˚ qq ñ H m ` q p Tot p X ˚˚ qq . The latter together with ( B ) imply the proposition. (cid:3) Theorem 4.3.
Let p H, G, µ q be a crossed module acting on an abelian crossedmodule p C, A, ν q . Then, for a short exact sequence of crossed modules Ñ p H , G , µ q Ñ p H, G, µ q Ñ p H , G , µ q Ñ , there is a first quadrant spectral sequence E pq “ H p ` Ner ˚ p H , G , µ q , H q p Ner ˚ p H , G , µ q , Ner ˚ p C, A, ν qq ˘ ñ H p ` q pp H, G, µ q , p C, A, ν qq . Proof.
Straightforward from the previous proposition. (cid:3)
Lemma 4.4.
Let p α, β q : p N, G, σ q Ñ p N , G , σ q be a morphism of inclusioncrossed modules such that the induced map G { N Ñ G { N is an isomorphism.Suppose Ner ˚ p N , G , σ q acts on an abelian simplicial group A ˚ . Then there is anisomorphism H n p Ner ˚ p N, G, σ q , A ˚ q – H n p Ner ˚ p N , G , σ q , A ˚ q , n ě , where Ner ˚ p N, G, σ q acts on A ˚ via Ner ˚ p N, G, σ q Ñ
Ner ˚ p N , G , σ q . COMONADIC INTERPRETATION OF BAUES–ELLIS HOMOLOGY 11
Proof.
The morphism p α, β q : p N, G, σ q Ñ p N , G , σ q induces a homomorphism ofbicomplexes B p Ner ˚ p N, G, σ q , A ˚ q Ñ B p Ner ˚ p N , G , σ q , A ˚ q . By Lemma 3.9 wehave π n p B m p Ner ˚ p N, G, σ q , A ˚ qq “ B m ´ p G { N q b π n p A ˚ q ,π n p B m p Ner ˚ p N , G , σ q , A ˚ qq “ B m ´ p G { N q b π n p A ˚ q for m , n ě . Hence a spectral sequence argument implies that Tot B p Ner ˚ p N, G, σ q , A ˚ q Ñ Tot B p Ner ˚ p N , G , σ q , A ˚ q is a quasi-isomorphism. (cid:3) Recall that a morphism of crossed modules p α, β q : p H, G, µ q Ñ p H , G , µ q issaid to be a weak equivalence when the induced morphisms α : Ker µ Ñ Ker µ and β : G { Im µ Ñ G { Im µ are isomorphisms. As explained in [21], these weakequivalences are actually part of a Quillen model category structure on XMod ; inparticular, any homotopy equivalence of crossed modules is a weak equivalence.The following proposition shows that the homology groups of crossed modules arehomotopical invariants.
Proposition 4.5.
Let p α, β q : p H, G, µ q Ñ p H , G , µ q be a weak equivalence ofcrossed modules. Suppose p H , G , µ q acts on an abelian crossed module p C, A, ν q and consider the induced action of p H, G, µ q on p C, A, ν q . Then there is an iso-morphism H n pp H, G, µ q , p C, A, ν qq – H n pp H , G , µ q , p C, A, ν qq , n ě . Proof.
We have the following exact sequences of crossed modules: Ñ p
Ker µ, , q Ñ p H, G, µ q Ñ p H { Ker µ, G, µ q Ñ , Ñ p
Ker µ , , q Ñ p H , G , µ q Ñ p H { Ker µ , G , µ q Ñ . We define a bicomplex X ˚˚ (respectively X ) as in Proposition 4.2 by setting Γ ˚ “ Ner ˚ p Ker µ, , q , Π ˚ “ Ner ˚ p H, G, µ q and G ˚ “ Ner ˚ p H { Ker µ, G, µ q (respectively Γ “ Ner ˚ p Ker µ , , q , Π “ Ner ˚ p H , G , µ q and G “ Ner ˚ p H { Ker µ , G , µ q ).Then, the morphism p α, β q : p H, G, µ q Ñ p H , G , µ q induces a homomorphism ofbicomplexes X ˚˚ Ñ X . This homomorphism is compatible with filtration definedin Proposition 4.2. We have the following spectral sequences: E pq “ H p ` Ner ˚ p H { Ker µ, G, µ q , H q p Ner ˚ p Ker µ, , q , Ner ˚ p C, A, ν qq ˘ ñ H p ` q p Tot p X ˚˚ qq “ H p ` q pp H, G, µ q , p C, A, ν qq ,E pq “ H p ` Ner ˚ p H { Ker µ , G , µ q , H q p Ner ˚ p Ker µ , , q , Ner ˚ p C, A, ν qq ˘ ñ H p ` q p Tot p X qq “ H p ` q pp H , G , µ q , p C, A, ν qq . Since both p H { Ker µ, G, µ q and p H { Ker µ , G , µ q are inclusion crossed modules,Lemma 4.4 yields an isomorphism E pq – E pq for each p ě , q ě . This impliesthat Tot p X ˚˚ q and Tot p X q are quasi-isomorphic. (cid:3) Another consequence of the Lyndon–Hochschild–Serre spectral sequence is a five-term exact sequence relating homologies of crossed modules in lower dimensions.Recall that for each crossed module p H, G, µ q its abelianisation is given by p H, G, µ q ab “ p H {r G, H s , G {r G, G s , µ q , where r G, H s ⊳ H is the normal subgroup of H generated by all g hh ´ for g P G , h P H (we refer the reader to [10] to see that this is correct from the categoricalpoint of view). It is easy to see that if p H, G, µ q acts trivially on p , Z , q then H p Ner ˚ p H, G, µ q , Ner ˚ p , Z , qq “ Ner ˚ p H, G, µ q ab . ( C ) Proposition 4.6.
Given a short exact sequence of crossed modules Ñ p H , G , µ q Ñ p H, G, µ q Ñ p H , G , µ q Ñ , we have the following exact sequence: H p G, H, µ q Ñ H p H , G , µ q Ñ G {p µ p H qr G, G sqÑ G {p µ p H qr G, G sq Ñ G {p µ p H qr G , G sq Ñ Moreover, there is a homomorphism H p H, G, µ q Ñ H pp H , G , µ q , p H , G , µ q ab q and an epimorphism from Ker ` H p G, H, µ q Ñ H p G , H , µ q ˘ to Coker ` H p H, G, µ q Ñ H pp H , G , µ q , p H , G , µ q ab q ˘ . Proof.
Suppose that p H, G, µ q acts trivially on p , Z , q . Then the Lyndon–Hoch-schild–Serre spectral sequence assumes the following form: E pq “ H p ` Ner ˚ p H , G , µ q , H q p Ner ˚ p H , G , µ q , Ner ˚ p , Z , qq ˘ ñ H p ` q p H, G, µ q . Since this is a first quadrant spectral sequence, we have the following exact sequence: H p H, G, µ q Ñ E Ñ E Ñ H p H, G, µ q Ñ E Ñ . Clearly E “ H p H , G , µ q and E “ H p H , G , µ q . By Proposition 3.5,Proposition 3.6 and ( C ) we have: E “ H pp H , G , µ q , Ner ˚ p H , G , µ q ab q “ G {p µ p H qr G, G sq ,H p H, G, µ q “ G {p µ p H qr G, G sq ,E “ H p H , G , µ q “ G {p µ p H qr G , G sq . This implies the first part of the proposition.Furthermore, since E pq is a first quadrant spectral sequence, we have an epi-morphism: Ker ` H p H, G, µ q Ñ E ˘ Ñ E , and E “ Coker ` E Ñ E ˘ . Substituting E and E for the suitable homology groups, we obtain the desiredresult. (cid:3) Given a crossed module p H, G, µ q , a free presentation of p H, G, µ q is definedto be a short exact sequence of crossed modules Ñ p R , F , η q Ñ p R, F, η q Ñ p
H, G, µ q Ñ , ( D )where p R, F, η q is a free crossed module (with respect to Set ). Corollary 4.7 (Hopf Formula) . For any free presentation ( D ) , there is an iso-morphism: H p H, G, µ q – p η p R q ¨ r F, F sq X F η p R q ¨ r F, F s . Proof.
Straightforward from Proposition 4.6 and Corollary 3.11. (cid:3)
Corollary 4.8.
For any crossed module p H, G, µ q , there exists an epimorphism H p H, G, µ q ։ H p G { µ p H qq . COMONADIC INTERPRETATION OF BAUES–ELLIS HOMOLOGY 13
Proof.
We have the following short exact sequence of crossed modules: Ñ p
Ker µ, , q Ñ p H, G, µ q Ñ p H { Ker µ, G, µ q Ñ . This implies the existence of an epimorphism H p H, G, µ q ։ H p H { Ker µ, G, µ q .Moreover, since p H { Ker µ, G, µ q is an inclusion crossed module, H p H { Ker µ, G, µ q is isomorphic to H p G { µ p H qq . (cid:3) In Section 6 we shall consider higher-order versions of these two results.
Proposition 4.9.
For any free presentation ( D ) , there is an isomorphism H p H, G, µ q – H pp H, G, µ q , p R , F , η q ab q . Proof.
By Corollary 3.11 we have H n p R, F, η q “ , n ě . Therefore, from theLyndon–Hochschild–Serre spectral sequence, E pq “ H p ` Ner ˚ p H, G, µ q , H q p Ner ˚ p R , F , η q , Ner ˚ p , Z , qq ˘ ñ H p ` q p R, F, η q , we derive that E Ñ E is an epimorphism and Ker p E Ñ E q must be iso-morphic to E . Since E “ H p H, G, µ q and E “ H pp H, G, µ q , p R , F , η q ab q , itsuffices to show that E “ . By the definition H q p Ner ˚ p R , F , η q , Ner ˚ p , Z , qq has the following form: ¨ ¨ ¨ , , , , H m p R ¸ R ¸ F , Z q , , , H m p R ¸ F , Z q , , H q p F , Z q . Since F is a free group, H q p F , Z q “ , q ě . Consequently E q “ H ` Ner ˚ p H, G, µ q , H q p Ner ˚ p R , F , η q , Ner ˚ p , Z , qq ˘ “ for all q ě . (cid:3) Homology via non-abelian left derived functors
We come back to the description of the forgetful/free adjunction between
XMod and
Set recalled just above Corollary 3.11. The pair of adjoint functors p F, U q in-duces a comonad P “ p P, δ, ǫ q on XMod in the usual way: P “ F U : XMod Ñ XMod , ǫ : P Ñ XMod is the counit and δ “ F ηU : P Ñ P where η : 1 Set Ñ U F is theunit of the adjunction. Given any crossed module p H, G, µ q , there is an augmentedsimplicial object P ˚ p H, G, µ q Ñ p
H, G, µ q in the category XMod where P n p H, G, µ q “ P n ` p H, G, µ q ,d ni “ P i p ǫ p P n ´ i p H, G, µ qqq , s ni “ P i p δ p P n ´ i p H, G, µ qqq , ď i ď n. This is called the P -cotriple resolution of p H, G, µ q .Let Ab denote the category of abelian groups and let T : XMod Ñ Ab be afunctor. As in [1], the left derived functors of T with respect to the comonad P are given, for any crossed module p H, G, µ q , by L n T p H, G, µ q “ π n p T p P ˚ p H, G, µ qqq , where T p P ˚ p H, G, µ qq is the simplicial abelian group obtained by applying the func-tor T dimension-wise to the P -cotriple resolution of p H, G, µ q . Note that the homo-topy groups π n p T p P ˚ p H, G, µ qqq agree with the homology groups of the simplicialabelian group T p P ˚ p H, G, µ qq . The functors L n T , for n ě , may also be inter-preted as non-abelian left derived functors , in the sense of [25], of T relative to theprojective class P determined by the comonad P .Let A : XMod Ñ Ab be the functor given by p H, G, µ q ÞÑ A p H, G, µ q “ G {p µ p H q ¨ r G, G sq . ( E ) Our goal is to show that L n A p H, G, µ q is isomorphic to H n ` p H, G, µ q for each n ě . Since the functor A : XMod Ñ Ab is right exact, by Proposition 3.6 we have: L A p H, G, µ q “ A p H, G, µ q “ G {p µ p H qr G, G sq “ H p H, G, µ q . Lemma 5.1.
For each n ě , the following augmented simplicial group is aspher-ical: ¨ ¨ ¨ , , , Ner n p P p H, G, µ qq , , Ner n p P p H, G, µ qq , Ner n p H, G, µ q . Here
Ner ˚ p P i p H, G, µ qq is the nerve of the crossed module P i p H, G, µ q .Proof. By the definition each P i p H, G, µ q , i ě , is an inclusion crossed module.Thus we may write P i p H, G, µ q “ p R i , F i , η q , i ě , where the R i and F i are free.Then, both ¨ ¨ ¨ , , , , F , , , F , , F , G and ¨ ¨ ¨ , , , , R , , , R , , R , H are aspherical augmented simplicial groups (see [10]). This implies the lemma. (cid:3) Theorem 5.2.
There is an isomorphism L n A p H, G, µ q – H n ` p H, G, µ q , n ě . Proof.
Consider the bicomplex B (cid:12) (cid:18) ´B (cid:12) (cid:18) B (cid:12) (cid:18) X B (cid:12) (cid:18) X ´B (cid:12) (cid:18) l r X B (cid:12) (cid:18) l r ¨ ¨ ¨ l r X X l r X l r ¨ ¨ ¨ , l r where p X p ˚ , Bq “
Tot B ` Ner ˚ p P p p H, G, µ qq , Ner ˚ p , Z , q ˘ . Lemma 3.9 togetherwith Lemma 5.1 imply that Tot X ˚˚ and Tot B ` Ner ˚ p H, G, µ q , Ner ˚ p , Z , q ˘ arequasi-isomorphic. As a consequence, H n p Tot X ˚˚ q “ H n p H, G, µ q , n ě . ( F )On the other hand we have the following spectral sequence: E pq “ H q p X p ˚ q “ H q p P p p H, G, µ qq ñ H p ` q p Tot X ˚˚ q . Since P p p H, G, µ q is a free crossed module for all p ě , by Proposition 3.5, Pro-position 3.6 and Corollary 3.11 we have: E pq “ $’&’% Z when q “ , A p P p p H, G, µ qq when q “ , when q ě .This implies that E pq “ $’’’&’’’% Z when p “ and q “ , when p ą and q “ , π p p A p P ˚ p H, G, µ qqq when q “ , when q ě .Since E pq “ E pq , we derive an isomorphism: H n ` p Tot X ˚˚ q – π n p A p P ˚ p H, G, µ qqq – L n A p H, G, µ q , n ě . ( G )Thus, ( F ) and ( G ) complete the proof. (cid:3) COMONADIC INTERPRETATION OF BAUES–ELLIS HOMOLOGY 15 Higher Hopf formulae
Theorem 5.2 may be used to obtain Hopf formulae for H n ` p H, G, µ q . Indeed,the functor A is of a type considered in [17], so that the general theory developedthere applies.Given a crossed module p H, G, µ q , let P ˚ p H, G, µ q “ p R ˚ , F ˚ , η q be its P -cotripleresolution. Let us fix some n ě , and write x n y “ t , . . . , n ´ u . Following [18, 16],the n -truncation p R i , F i , η q i ă n of this resolution may be considered as an n -foldfree presentation of p H, G, µ q . We shall write p R, F, η q for p R n ´ , F n ´ , η q , anddenote the kernel of d ni : p R n ´ , F n ´ , η q Ñ p R n ´ , F n ´ , η q for i P x n y by p R i , F i , η i q ⊳ p R, F, η q . Then we find the following: Theorem 6.1 (Hopf Formula) . There is an isomorphism H n ` p H, G, µ q – p η p R q ¨ r F, F sq X Ş i P n F i η p Ş i Px n y R i q ¨ ś I Ďx n y r Ş i P I F i , Ş i R I F i s . Proof.
This is an application of Corollary 6.12 in [17]. In fact the situation we areconsidering is an instance of an example given on page 3674 of that paper, in thesubsection entitled
Internal groupoids with coefficients in abelian objects . Via theequivalence between internal categories and crossed modules, it is the special caseof that example for A “ Gp .The functor A : XMod Ñ Ab : p H, G, µ q ÞÑ G {p µ p H q ¨ r G, G sq may thus be seen asthe reflector of XMod to an intersection of two of its subvarieties: Ab p XMod q on theone hand, and Gp (discrete crossed modules) on the other. The reflector to Gp is pro-toadditive , being the connected components functor viewed as p H, G, µ q ÞÑ G { µ p H q .Hence the requirements of [17, Corollary 6.12] are satisfied. (cid:3) Note how this particularises to Corollary 4.7 when n “ . Similarly, the firstpart of Proposition 4.6 follows from a result in [19] via the interpretation in termsof cotriple homology. In fact, it is the tail of a long exact sequence: see [15, 22]. References [1] M. Barr and J. Beck,
Homology and standard constructions , Seminar on triples and categor-ical homology theory (ETH, Zürich, 1966/67), Lecture Notes in Math., vol. 80, Springer,1969, pp. 245–335.[2] H. J. Baues,
Combinatorial homotopy and -dimensional complexes , De Gruyter, Berlin,1991.[3] J. M. Beck, Triples, algebras and cohomology , Reprints in Theory and Applications of Cat-egories (2003), 1–59, Ph.D. thesis, Columbia University, 1967.[4] F. Borceux, G. Janelidze, and G. M. Kelly, Internal object actions , Comment. Math. Univ.Carolinae (2005), no. 2, 235–255.[5] D. Bourn and G. Janelidze, Protomodularity, descent, and semidirect products , Theory Appl.Categ. (1998), no. 2, 37–46.[6] L. Breen, Théorie de Schreier supérieure , Ann. Sci. Éc. Norm. Supér. (4) (1992), no. 5,465–514.[7] R. Brown and G. J. Ellis, Hopf formulae for the higher homology of a group , Bull. Lond.Math. Soc. (1988), no. 2, 124–128.[8] R. Brown and J. Huebschmann, Identities among relations , Low-Dimensional Topology(Cambridge), London Math. Soc. Lecture Note Ser., vol. 46, Cambridge Univ. Press, 1982,pp. 153–202.[9] R. Brown and J.-L. Loday,
Van Kampen theorems for diagrams of spaces , Topology (1987), no. 3, 311–335.[10] P. Carrasco, A. M. Cegarra, and A. R.-Grandjeán, (Co)Homology of crossed modules , J. PureAppl. Algebra (2002), no. 2-3, 147–176.[11] J. M. Casas, N. Inassaridze, E. Khmaladze, and M. Ladra, Adjunction between crossed mod-ules of groups and algebras , J. Homotopy Relat. Struct. (2014), 223–237. [12] A. S. Cigoli, J. R. A. Gray, and T. Van der Linden, Algebraically coherent categories , TheoryAppl. Categ. (2015), no. 54, 1864–1905.[13] G. Donadze, N. Inassaridze, and T. Porter, n -Fold Čech derived functors and generalisedHopf type formulas , K-Theory (2005), no. 3–4, 341–373.[14] G. J. Ellis, Homology of -types , J. London Math. Soc. (2) (1992), 1–27.[15] T. Everaert, An approach to non-abelian homology based on Categorical Galois Theory , Ph.D.thesis, Vrije Universiteit Brussel, 2007.[16] T. Everaert, J. Goedecke, and T. Van der Linden,
Resolutions, higher extensions and therelative Mal’tsev axiom , J. Algebra (2012), 132–155.[17] T. Everaert and M. Gran,
Protoadditive functors, derived torsion theories and homology ,J. Pure Appl. Algebra (2015), no. 8, 3629–3676.[18] T. Everaert, M. Gran, and T. Van der Linden,
Higher Hopf formulae for homology via GaloisTheory , Adv. Math. (2008), no. 5, 2231–2267.[19] T. Everaert and T. Van der Linden,
Baer invariants in semi-abelian categories II: Homology ,Theory Appl. Categ. (2004), no. 4, 195–224.[20] M. Forrester-Barker, Representations of crossed modules and
Cat -groups , Ph.D. thesis, Uni-versity of Bangor, 2003.[21] A. R. Garzón and J. G. Miranda, Homotopy theory for (braided) Cat-groups , Cah. Topol.Géom. Differ. Catég.
XXXVIII (1997), no. 2, 99–139.[22] J. Goedecke and T. Van der Linden,
On satellites in semi-abelian categories: Homologywithout projectives , Math. Proc. Cambridge Philos. Soc. (2009), no. 3, 629–657.[23] A. R.-Grandjeán, M. Ladra, and T. Pirashvili,
CCG-homology of crossed modules via classi-fying spaces , J. Algebra (2000), 660–665.[24] P. J. Higgins,
Groups with multiple operators , Proc. Lond. Math. Soc. (3) (1956), no. 3,366–416.[25] H. N. Inassaridze, Non-abelian homological algebra and its applications , Kluwer Acad. Publ.,Dordrecht, 1997.[26] G. Janelidze, L. Márki, and W. Tholen,
Semi-abelian categories , J. Pure Appl. Algebra (2002), no. 2–3, 367–386.[27] R. Lavendhomme and J. R. Roisin,
Cohomologie non abélienne de structures algébriques ,J. Algebra (1980), 385–414.[28] S. Mac Lane, Homology , Grundlehren math. Wiss., vol. 114, Springer, 1967.[29] S. Mac Lane,
Categories for the working mathematician , second ed., Grad. Texts in Math.,vol. 5, Springer, 1998.[30] S. Mac Lane and J. H. C. Whitehead,
On the -type of a complex , Proc. Nat. Acad. Sci.U.S.A. (1950), 41–48.[31] N. Martins-Ferreira and T. Van der Linden, A note on the “Smith is Huq” condition , Appl.Categ. Structures (2012), no. 2, 175–187.[32] K. Norrie, Crossed modules and analogues of group theorems , Ph.D. thesis, King’s College,University of London, 1987.[33] K. Norrie,
Actions and automorphisms of crossed modules , Bull. Soc. Math. France (1990), 129–146.[34] Ch. A. Weibel,
An introduction to homological algebra , Cambridge Stud. Adv. Math., vol. 38,Cambridge Univ. Press, 1994.[35] J. H. C. Whitehead,
Combinatorial homotopy II , Bull. Amer. Math. Soc. (1949), 453–496.(Guram Donadze) Indian Institute of Science Education and Research (IISER-TRV),CET Campus, TVPM-16, Thiruvananthapuram, Kerala, India
E-mail address : [email protected] (Tim Van der Linden) Institut de Recherche en Mathématique et Physique, Uni-versité catholique de Louvain, chemin du cyclotron 2 bte L7.01.02, 1348 Louvain-la-Neuve, Belgium
E-mail address ::