Reach Us
+44-1522-440391

Medical, Pharma, Engineering, Science, Technology and Business

**Everaert T and Gran M ^{*}**

Catholic University of Louvain, Place de l'Université 1, 1348, Belgium

- Corresponding Author:
- Gran M

President of Research Institute of Mathematics and Physics

Catholic University of Louvain, Place de l'Université 1

1348, Belgium

**Tel:**+3210472111

**E-mail:**[email protected]

**Received date** October 11, 2015; **Accepted date** November 20, 2015; **Published date** November 27, 2015

**Citation:** Everaert T, Gran M (2015) Some Aspects of Semi-Abelian Homology and Protoadditive Functors. J Generalized Lie Theory Appl 9:238. doi:10.4172/1736-4337.1000238

**Copyright:** © 2015 Everaert T, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

In this note some recent developments in the study of homology in semi-abelian categories are briefly presented. In particular the role of protoadditive functors in the study of Hopf formulae for homology is explained.

**Semiabelian homology**; Centralisation functor; Protoadditive functors

The discovery of higher Hopf formulae for the homology of a group, due to Ronald Brown and Graham Ellis [1], has naturally led to some new perspectives in non-abelian homological algebra. An important advance in this area was made by George Janelidze, who found a connection between group homology and categorical Galois theory [2-5], the latter being a wide extension of Alexander Grothendieck’s theory [6]. Among other things, this cleared the path for the discovery of higher Hopf formulae for the homology of general algebraic structures [7-14]. Here, a crucial role is played by the so-called *higher order central extensions*, which are the covering morphisms with respect to certain Galois structures induced by a reflection

(1)

whose left adjoint is sometimes called the “coefficient functor”. Here, could, for instance, be the variety of groups, its subvariety of abelian groups, and I the abelianisation functor. In this case, the induced higher order central extensions are related to the Brown- Ellis Hopf-formulae, as explained below. More generally, higher order central extensions can be defined for any semi-abelian category ([15] e.g., the varieties of groups, rings, Lie algebras, (pre)crossed modules, **compact groups**, or any abelian category) and any Birkhoff subcategory (i.e., a reflective subcategory closed under subobjects and regular quotients) of . When, moreover, has enough projectives, one obtains higher Hopf formulae for the homology induced by the reflector (or, coefficient functor) .

In order to explicitly determine the Hopf-Brown-Ellis formulae for homology in specific algebraic contexts, it is crucial to find suitable descriptions of the higher central extensions, as for instance in terms of algebraic conditions using “generalised commutators”. In general, this is a non-trivial problem, about which we are going to say more in what follows.

Note that, in fact, different approaches to obtaining higher Hopf formulae exist, which can be used in different categorical contexts, based on the comonadic homology theory of Barr and Beck [7,8,10,16], on the abstract notion of Galois group [13,17,18], or on the theory of satellites [19]. These methods essentially coincide in the situation described above, namely for a semi-abelian category with enough projectives and a Birkhoff subcategory of .

Assume that we are in this situation. In this case, the reflector induces a first “**centralisation functor**” I_{1} from the category Ext () of extensions (i.e., regular epimorphisms) in to the full subcategory CExt () of extensions that are central with respect to .

(2)

This functor I_{1} is the left adjoint of the inclusion functor U_{1}. The notion of centrality comes from categorical Galois theory and depends on the choice of the Birkhoff subcategory . It is defined in purely categorical terms [10,20].

The centralisation I_{1}(f) of an extension f is given by a quotient

where [Ker(f), A] may be thought of as a commutator of Ker(f) and A, defined relatively to . For instance, in the classical case of the reflection

this relative commutator is simply the group-theoretical commutator of normal subgroups:

[Ker(f), A]_{Ab} = [Ker(f), A].

Hence, in this case I_{1} is the usual centralisation functor from the category of group extensions to its full subcategory of central extensions in the classical sense.

Remark that the commutator [Ker(f), A] appears as the denominator in the Hopf formula for the second integral homology group: given a free presentation

0 → K → F → G → 0,

of a group G, Hopf’s formula tells us that the second homology group is given by the quotient

This is not a coincidence: a similar phenomenon occurs for the higher-order homology groups, where the subgroups appearing as denominator of the Brown-Ellis Hopf formulae are exactly what is required to transform higher-order extensions into a higher-order *central* extensions, universally.

To illustrate this idea, consider the case of the *third* homology group. For this, let

be a double presentation of a group G so that F, *F/K _{1}* and

(4)

Once more, the denominator gives precisely the normal subgroup by which one has to “quotient out” the group *F* in order to make this double extension a double *central* extension, universally:

Once again, the notion of centrality comes from categorical Galois theory, and this time depends on the induced reflection (2).

Now, the formula (4) is a special case of the general Hopf formula for the third homology corresponding to a reflection (1), with an arbitrary semi-abelian category with enough projectives, and any Birkhoff subcategory of [10]: starting from a double presentation f of the form (3), with F, F/K_{1} and F/K_{2} regular projective objects of the third homology object is given by a quotient

(5)

As in the case of groups, also in a general semi-abelian category the denominator L_{2} [f] of this generalised Hopf formula relative to is the normal subobject of F that has to be “quotiented out” in order to universally turn the double extension (3) into a double central extension. Hence, in particular, for = Gp and = Ab, we have the equality

and the formula (4) appears as a special case of (5). In general, for a given Birkhoff reflection (1) in a semi-abelian category, L_{2}[ f] may be difficult to compute.

Similar formulae exist for the higher homology objects, again valid in any semi-abelian category with enough projectives and for any Birkhoff subcategory of . This yields, at least in principle, a description of all homology objects Hn(A,)(n ≥ 2). In practice, a suitable characterisation of the higher-order central extensions is required.

In some cases it has been possible to compute these formulae explicitly. For example, this has been done in literature of Everaert [10] for the variety of precrossed modules and its subvariety of crossed modules, or for the variety of groups and its subvariety of nilpotent groups of a fixed class k ≥ 1 [21], or the variety of solvable groups of a fixed class k ≥ 1. Similar results have been obtained in the categories of Leibniz and of Lie n-algebras in studies of Casas [22].

However, in general, computing the Hopf formulae explicitly is a non-trivial task. One possible strategy is to look for suitable conditions on the coefficient functor that facilitate such computations. In literature of Everaert [12,14] we have shown that a natural such condition is the requirement that the reflector I is a *protoadditive functor*. This notion extends the one of additive functor to the nonadditive context of pointed protomodular categories [23]: when and are pointed protomodular categories (for instance, and could be semi-abelian), a functor is *protoadditive* if, for any split short exact sequence

in (i.e., and k = Ker(f)), its image

under I is a split short exact sequence in . Whenever the coefficient functor is protoadditive, explicit Hopf formulae can be established in different algebraic and topological contexts. In particular, the protoadditivity condition is fundamental to explore some new Galois theories induced by torsion theories [24-26].

We refer the interested reader to the articles [12,14] for a thorough study of the theory of protoadditive functors and their use in semiabelian **homological algebra**. In the first article we study the homology of n-fold internal groupoids in a semi-abelian category: these results apply in particular to the so-called catn-groups in the sense of Loday [27]. The crucial point there is that the connected components functor π_{0}: Grpd() → is protoadditive whenever is semi-abelian. In studies of Everaert [14] the general theory of protoadditive functors is investigated, as well as the so-called *derived torsion theories* of a torsion theory having the reflector to its torsion-free subcategory *protoadditive*. These derived torsion theories induce a chain of Galois structures in the categories of higher order extensions. The results concerning the **homology** objects can be applied in particular to some new torsion theories in the category of compact groups and, more generally, in any category of compact semi-abelian algebras introduced in framework of Borceux and Clementino [28]. Further developments in this direction, including some new results in the categories of commutative rings and of topological groups, for instance, can be found in literature of Duckerts-Antoine [13,18].

This article is partly based on the text of a talk given by the second author on this joint work at the *Séminaire Itinérant des Catégories* that took place at the Université Paris Didérot on October 25, 2009.

- Brown R, Ellis G (1988)Hopf formulae for the higher homology of a group. Bull. Lond. Math. Soc. 20: 124-128.
- Janelidze G(1990) Pure Galois theory in categories. J. Algebra 132: 270-286.
- Janelidze G(1991) What is a double central extension? (The question was asked by Ronald Brown). Cah. Topologie Géom. Différ. Catég. 32: 191-201.
- Janelidze G(1995) Higher dimensional central extensions: A categorical approach to homology theory of groups. Lecture at the International Category Theory Meeting CT95.
- Borceux F, Janelidze G(2001) Galois theories. Cambridge University Press, Cambridge Studies in Adv. Mathematics 72.
- Grothendieck A (1971)Stalls coverings and fundamental group. SGA1 exposed V player. Notes in Math., Springer 224.
- Everaert T, Van der Linden T(2004) Baer invariants in semi-abelian categories I: General Theory. Theory Appl. Categ. 12: 1-33.
- Everaert T, Van der LindenT (2004)Baer invariants in semi-abelian categories II: Homology. Theory Appl. Categ. 12: 195-224.
- Everaert T, Gran M (2007) On low-dimensional homology in categories, Homology. Homotopy and Applications 9: 275-293.
- Everaert T, Gran M, Van der Linden T(2008) Higher Hopf Formulae for Homology via Galois Theory. Adv. Math. 217: 2231-2267.
- Everaert T(2010) Higher central extensions and Hopf formulae. J. Algebra 324: 1771-1789.
- Everaert T and Gran M (2010)Homology of
*n*-fold groupoids. Theory Appl. Categ. 23: 22-41. - Duckerts M, Everaert T, Gran M(2012) A description of the fundamental group in terms of commutators and closure operators. J. Pure Appl. Algebra 216: 1837-1851.
- Everaert T, Gran M (2015)Protoadditive functors, derived torsion theories and homology. J. Pure Appl. Algebra 219: 3629-3676.
- Janelidze G, Márki L, Tholen W(2002) Semi-abelian categories. J. Pure Appl. Algebra 168: 367-386.
- Barr M, Beck J(1969) Homology and standard constructions. Seminar on Triples and Categorical Homology Theory, Lectures Notes in Math 80: 245-335.
- Janelidze G (2008)Galois groups, abstract commutators and Hopf formula. Appl. Categ. Struct. 16: 653-668.
- Duckerts-Antoine M(2015) Fundamental group functors in descent-exact homological categories. Preprint 15-38.
- Goedecke J, Van der Linden T(2009) On satellites in semi-abelian categories: Homology without projectives. Math. Proc. Cambridge Philos. Soc. 147: 629-657.
- Janelidze G, Kelly GM (1994)Galois theory and a general notion of central extension. J. Pure Appl. Algebra 97: 135-161.
- Donadze G, Inassaridze N, Porter T(2005)
*n*-fold Cech derived functors and generalised Hopf type formulas. K-Theory 35: 341-373. - Casas JM, Khmaladze E, Ladra M, Van Der LindenT (2011) Homology and central extensions of Leibniz and Lie
*n*-algebras, Homology. Homotopy and Applications 13: 59-74. - BournD (1991) Normalization equivalence, kernel equivalence and affine categories. Lecture Notes Math 1488: 43-62.
- Bourn D, Gran M(2006) Torsion theories in homological categories. J. Algebra 305: 18-47.
- Gran M, JanelidzeG (2009) Normal coverings and monadic extensions of Galois structures associated with torsion theories. Cah. Topologie Géom. Différ. Catég. 3: 171-188.
- Gran M, Rossi V(2007) Torsion theories and Galois coverings of topological groups. J. Pure Appl. Algebra 208: 135-151.
- Loday JL(1982) Spaces with finitely many non-trivial homotopy groups. J. Pure Appl. Algebra 24: 179-202.
- Borceux F, Clementino MM (2005)Topological semi-abelian algebras. Adv. in Math. 190: 425-453.

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebra
- Algebraic Geometry
- Analytical Geometry
- Applied Mathematics
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Binary and Non-normal Continuous Data
- Binomial Regression
- Biometrics
- Biostatistics methods
- Clinical Trail
- Combinatorics
- Complex Analysis
- Computational Model
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Deformations Theory
- Differential Equations
- Differential Transform Method
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Geometry
- Hamilton Mechanics
- Harmonic Analysis
- Homological Algebra
- Homotopical Algebra
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Latin Squares
- Lie Algebra
- Lie Superalgebra
- Lie Theory
- Lie Triple Systems
- Loop Algebra
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Operad Theory
- Physical Mathematics
- Quantum Group
- Quantum Mechanics
- Quantum electrodynamics
- Quasi-Group
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Representation theory
- Riemannian Geometry
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Super Algebras
- Symmetric Spaces
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topologies
- Topology
- mirror symmetry
- vector bundle

- Total views:
**8671** - [From(publication date):

December-2015 - Dec 17, 2018] - Breakdown by view type
- HTML page views :
**8463** - PDF downloads :
**208**

Peer Reviewed Journals

International Conferences 2018-19