## Higher Hopf formulae for homology via Galois Theory (2007)

Citations: | 11 - 8 self |

### BibTeX

@MISC{Everaert07higherhopf,

author = {Tomas Everaert and Marino Gran and Tim Van Der Linden},

title = {Higher Hopf formulae for homology via Galois Theory },

year = {2007}

}

### OpenURL

### Abstract

We use Janelidze’s Categorical Galois Theory to extend Brown and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A

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Citation Context ...c = cc c for all c, c ′ ∈ C. We will sometimes abbreviate the notation (C, G, ∂) to C. It is well known that the category of precrossed modules is equivalent to a variety of Ω-groups (see, e.g., [31] =-=[32]-=- or [29]). Via this equivalence, XMod correspond to a subvariety of PXMod. Hence, we can consider the Hopf formulae 8.1 in the case where A = PXMod and B = XMod. Recall that a precrossed submodule of ... |

40 | Semi-abelian categories
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Citation Context ...lso holds in categories of higher extensions. That such an approach is possible is due to the existence of the appropriate categorical framework, Janelidze, Márki and Tholen’s semi-abelian categories =-=[29]-=- and Borceux and Bourn’s homological categories [3]. These were introduced to capture the fundamental homological properties of the categories of groups, rings, Lie algebras, crossed modules etc. much... |

37 |
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27 |
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Citation Context ...is given of the Hopf formula for the second homology object, and some of the needed homological tools are developed. A categorical theory of central extensions was developed by Janelidze and Kelly in =-=[28]-=- as an application of Janelidze’s Categorical Galois Theory [24]. This theory is modelled on the situation where A is a variety of universal algebras and B a given subvariety of A, and allows one to c... |

26 |
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Citation Context ... some of the needed homological tools are developed. A categorical theory of central extensions was developed by Janelidze and Kelly in [28] as an application of Janelidze’s Categorical Galois Theory =-=[24]-=-. This theory is modelled on the situation where A is a variety of universal algebras and B a given subvariety of A, and allows one to classify the extensions in A that are central with respect to thi... |

24 | Some remarks on Maltsev and Goursat categories - Carboni, Kelly, et al. - 1993 |

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Citation Context ...t to centralization 21 7. The Hopf formula for the second homology object 22 8. The higher Hopf formulae 24 9. Examples 27 Index of notations 34 References 34 Introduction Generalizing Hopf’s formula =-=[23]-=- for the second integral homology group to higher dimensions is a well-studied problem, that still deserves to be better understood from a categorical perspective. Partial results were originally obta... |

20 |
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Citation Context .... It has an augmentation ǫA: GA −→ A; the augmented simplicial object ǫA: GA −→ A is called the canonical G-simplicial resolution of A. The following naturally generalizes Barr-Beck cotriple homology =-=[2]-=- to the semi-abelian context. Definition 5.1. [18] Let A be a category equipped with a comonad G and B a semi-abelian category. Let I : A −→ B be a functor. For n ≥ 1, the object Hn(A, I)G = Hn−1NIGA ... |

20 |
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Citation Context ...r the study of higher central extensions. In particular we give a useful sufficient condition for a Galois structure to be admissible in the sense of this theory. We refer the reader to the monograph =-=[4]-=- by Borceux and Janelidze, and in particular to its introduction, for the historical background that led to the development of the theory, as well as for the details of several interesting examples of... |

19 |
Homotopy algebra, Lecture notes in mathematics
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Citation Context ...o its Birkhoff subcategory of abelian groups, and G the underlying set/free group comonad, it is well known that Hn(A, ab)G is just the n-th integral homology group Hn(A, Z) of A (see page II.6.16 of =-=[34]-=-).HIGHER HOPF FORMULAE FOR HOMOLOGY VIA GALOIS THEORY 19 5.2. Comonads derived from a suitable adjunction. From now on we shall assume that A is semi-abelian. We are going to construct simplicial res... |

13 | Protomodularity, descent and semi-direct products
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(Show Context)
Citation Context ... in Ext n−1 A. We are going to show that Jn−1π ′ 1 = Jn−1π ′ 2 . Since Jn−1π ′ 1 and Jn−1π ′ 2 are jointly monic, this implies that Jn−1π ′ 1 is a monomorphism hence an iso, so that by Theorem 2.3 in =-=[9]-=-, (ii) is a pullback. Since CExt n−1 B A is a strongly En−1-Birkhoff subcategory of Ext n−1 A, Jn−1 preserves n-extensions, hence the left hand downward pointing arrow in the diagram Jn−1R[f] Jn−1π1 J... |

13 |
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Citation Context ...f] : Jn+1[f] H2(A, In)Gn is the direct image of K[f] ∩ JnB along ρ n+1 f : B −→ In+1[f]. □ Remark 7.4. In particular, the expressions on the right hand side of these Hopf formulae are Baer invariants =-=[19, 17]-=-: they are independent of the chosen presentation of A. Remark 7.5. Note that the objects in this Hopf formula are in Arr n A. Notation 7.6. The functor category Arr n A = Hom(2n , A) may with advanta... |

12 |
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Citation Context ...long f and dually, a cokernel Cokerf : A −→ Cok[f] is a pushout of B −→ 0 along f. A being Barr exact means that it is regular, and such that every internal equivalence relation in A is a kernel pair =-=[1]-=-. We start by commenting on the regularity, and later come back to the other condition. Recall that a morphism is called a regular epimorphism when it is a coequalizer of some pair of arrows. Having f... |

12 | Hopf formulae for the higher homology of a group
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(Show Context)
Citation Context ...). Both would like to thank the LMPA for its kind hospitality during their stay in Calais. 1� � 2 TOMAS EVERAERT, MARINO GRAN AND TIM VAN DER LINDEN describing Hn for all n—is due to Brown and Ellis =-=[11]-=-. Their work was recently extended by Donadze, Inassaridze and Porter in the paper [15]. Whereas Brown and Ellis use topological methods, the latter proof is entirely algebraic, and also considers the... |

12 |
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Citation Context ...irkhoff property of B. On the other hand, HA reflects isomorphisms since, again by the strongly E-Birkhoff property, ηA is a pullback-stable regular epimorphism: see, for instance, Proposition 1.6 in =-=[30]-=-. To see that these facts indeed imply that ǫA x is an isomorphism, for any x: X −→ IA in Z(IA), consider the triangular identity (HAǫA x )◦ηA HAx = 1HAx. Now, ηA HAx is both a split monomorphism and ... |

12 |
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Citation Context ...and allows one to classify the extensions in A that are central with respect to this chosen subvariety B. The idea of relative centrality—which goes back to the work of the Fröhlich school, see e.g., =-=[19, 33, 20]-=-—has for leading example the variety of groups with its subvariety of abelian groups. Janelidze and Kelly’s theory is general enough to include the case where A is any semi-abelian category and B is a... |

9 |
Baer invariants in semi-abelian categories
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Citation Context ...f] : Jn+1[f] H2(A, In)Gn is the direct image of K[f] ∩ JnB along ρ n+1 f : B −→ In+1[f]. □ Remark 7.4. In particular, the expressions on the right hand side of these Hopf formulae are Baer invariants =-=[19, 17]-=-: they are independent of the chosen presentation of A. Remark 7.5. Note that the objects in this Hopf formula are in Arr n A. Notation 7.6. The functor category Arr n A = Hom(2n , A) may with advanta... |

9 |
Homology and generalized Baer invariants
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Citation Context ...and allows one to classify the extensions in A that are central with respect to this chosen subvariety B. The idea of relative centrality—which goes back to the work of the Fröhlich school, see e.g., =-=[19, 33, 20]-=-—has for leading example the variety of groups with its subvariety of abelian groups. Janelidze and Kelly’s theory is general enough to include the case where A is any semi-abelian category and B is a... |

5 |
Galois theory and double central extensions
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Citation Context ...t the double extensions that are central with respect to this Galois structure Γ1 are precisely those extensions C with the property that [K[f0], K[b]] = {1} and [K[f0] ∩ K[b], B0] = {1}. Recently in =-=[22]-=- a similar characterization was obtained, valid in the context of Mal’tsev varieties. Notation 4.9. Let f : B −→ A be an n-extension. Recall from the definition of the Jn that Jnf = ΨJn[f]. It is easi... |

5 |
Cohomologie non abélienne de structures algébriques
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- 1980
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Citation Context ...′ −1 c = cc c for all c, c ′ ∈ C. We will sometimes abbreviate the notation (C, G, ∂) to C. It is well known that the category of precrossed modules is equivalent to a variety of Ω-groups (see, e.g., =-=[31]-=- [32] or [29]). Via this equivalence, XMod correspond to a subvariety of PXMod. Hence, we can consider the Hopf formulae 8.1 in the case where A = PXMod and B = XMod. Recall that a precrossed submodul... |

4 |
categories and fibration of pointed objects
- Mal’cev
- 1996
(Show Context)
Citation Context ...equivalence relation. For instance, if A is a Mal’tsev variety, then the subvarieties of A are strongly (regular epi)-Birkhoff subcategories. Recall that every semi-abelian category is exact Mal’tsev =-=[6, 3]-=-. In general, a Galois structure with the property that the adjunction satisfies the strongly E-Birkhoff property is always admissible: Proposition 2.6. Let Γ = (A, B, E, Z, I, H) be a Galois structur... |

4 | n-Fold Čech derived functors and generalised Hopf type formulas, K-Theory 35
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Citation Context ...is. 1� � 2 TOMAS EVERAERT, MARINO GRAN AND TIM VAN DER LINDEN describing Hn for all n—is due to Brown and Ellis [11]. Their work was recently extended by Donadze, Inassaridze and Porter in the paper =-=[15]-=-. Whereas Brown and Ellis use topological methods, the latter proof is entirely algebraic, and also considers the case of groups vs. k-nilpotent groups instead of just groups vs. abelian groups. The a... |

4 |
An approach to non-abelian homology based on Categorical Galois Theory
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- 2007
(Show Context)
Citation Context ... are pushouts. When A is a semi-abelian category, the category Arr n A is of course semi-abelian, as is any category of A-valued presheaves. On the other hand, while Reg n A is still homological (see =-=[16]-=-), it is no longer semi-abelian. Indeed, it is well known that for a (non-trivial) abelian category A the category RegA is not abelian; hence it cannot be exact, since it is obviously additive. To see... |

4 |
invariants in semi-abelian categories II
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Citation Context ...7]. As far as we know, our paper is the first attempt to use this idea for proving the formulae. Next to the concept of semiabelian categories, part of the fundamental theory is provided by the paper =-=[18]-=-, where a proof along the same lines is given of the Hopf formula for the second homology object, and some of the needed homological tools are developed. A categorical theory of central extensions was... |

3 |
A generalized Hopf formula for higher homology groups
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Citation Context ...igher dimensions is a well-studied problem, that still deserves to be better understood from a categorical perspective. Partial results were originally obtained by Conrad [14], Rodicio [35] and Stöhr =-=[36]-=-, and the first complete solution—a formula 2000 Mathematics Subject Classification. Primary 18G, 20J, 55N35, 18E10. Key words and phrases. Semi-abelian category, Hopf formula, homology, Galois theory... |

2 |
Crossed n-fold extensions of groups, n-fold extensions of modules, and higher multipliers
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Citation Context ... integral homology group to higher dimensions is a well-studied problem, that still deserves to be better understood from a categorical perspective. Partial results were originally obtained by Conrad =-=[14]-=-, Rodicio [35] and Stöhr [36], and the first complete solution—a formula 2000 Mathematics Subject Classification. Primary 18G, 20J, 55N35, 18E10. Key words and phrases. Semi-abelian category, Hopf for... |

2 |
is a double central extension? (The question was asked by Ronald
- What
- 1991
(Show Context)
Citation Context ...fulfils this promise. An important ingredient to understanding the higher Hopf formulae is Janelidze’s insight that centralization of higher extensions yields the objects that occur in these formulae =-=[26, 27]-=-. For instance, consider a group A presented by a double extension f : q −→ p F q � � F/K1 � �� F/K2 p � � � �� A where K1 and K2 are normal subgroups of F satisfying A ∼ = F/K1 · K2 and such that F, ... |

2 |
dimensional central extensions: A categorical approach to homology theory of groups, Lecture at the International Category Theory Meeting CT95
- Higher
- 1995
(Show Context)
Citation Context ...fulfils this promise. An important ingredient to understanding the higher Hopf formulae is Janelidze’s insight that centralization of higher extensions yields the objects that occur in these formulae =-=[26, 27]-=-. For instance, consider a group A presented by a double extension f : q −→ p F q � � F/K1 � �� F/K2 p � � � �� A where K1 and K2 are normal subgroups of F satisfying A ∼ = F/K1 · K2 and such that F, ... |

2 |
Presentaciones libres y H2n(G), Publ
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(Show Context)
Citation Context ...logy group to higher dimensions is a well-studied problem, that still deserves to be better understood from a categorical perspective. Partial results were originally obtained by Conrad [14], Rodicio =-=[35]-=- and Stöhr [36], and the first complete solution—a formula 2000 Mathematics Subject Classification. Primary 18G, 20J, 55N35, 18E10. Key words and phrases. Semi-abelian category, Hopf formula, homology... |

1 | µ n , 14 NExt n - Ln |

1 |
Semi-abelian monadic categories, Theory Appl. Categ
- Gran, Rosick´y
(Show Context)
Citation Context ...iety of algebras is monadic over Set, and thus semi-abelian varieties form an important class of examples. A characterization of such varieties is given by Bourn and Janelidze in their paper [10]. In =-=[21]-=-, Gran and Rosick´y characterize semi-abelian categories, monadic over Set. Definition 5.4. An object of Arr n A is called extension-projective when it is projective with respect to the class E n of (... |

1 |
and Galois theory, in Carboni et al
- Precategories
(Show Context)
Citation Context ... its introduction, for the historical background that led to the development of the theory, as well as for the details of several interesting examples of admissible Galois structures. Definition 2.1. =-=[25]-=- A Galois structure Γ = (A, B, E, Z, I, H) consists of two categories, A and B, an adjunction A �� I H ⊥ and classes E and Z of morphisms of A and B respectively, such that: (1) A has pullbacks along ... |