Dominique Bourn

Central extensions in semi-abelian categories

In the context of semi-abelian categories, we develop some new aspects of the categorical theory of central extensions by Janelidze and Kelly. If W is a semi-abelian category and X is any admissible subcategory we give several characterizations of trivial and central extensions. The notion of central extension becomes intrinsic when X is the subcategory of the abelian objects in W. We apply these results to the category of internal groupoids in a semi-abelian category. As a very special case, we get the known description of central extensions for crossed modules

Aspherical abelian groupoids and their directions

Abstract From the intrinsic notion of normal subobject and abelian object in a protomodular category, the notion of abelian groupoid in a category E is introduced. A cofibration d 1 , the direction functor, is built up from the category of aspherical abelian groupoid in E to the category of abelian groups in E . The fibres of d 1 are automatically endowed with a symmetric tensor product. The associated abelian group structure on the set of connected components of the fibre above the internal abelian group A realizes the second cohomology group H 2 ( E ,A) .

The tower of n-groupoids and the long cohomology sequence

Given an exact category E, we associate to it a fibration c above E such that, for each object X of E, the fiber c[x] is again exact. If, moreover, A is an internal abelian group in E, it determines a family of abelian groups Ax in the fibres c[x], such that the group H1(E,A) is the colimit of the H0(c[x],Ax). This remark allows us to define iteratively Hn+1(E,A) as the colimit of the Hn(c[x],Ax). These groups are shown to have the property of the long cohomology sequence. When E=Ab, the construction coincides, up to isomorphism, with Yonedas classical description of Extn. When E=Grp, it coincides with the cohomology groups of a group in the sense of Eilenberg-Mac Lane.

Extensions with Abelian Kernels in Protomodular Categories

Abstract As observed by J. Beck, and as we know from M. Barrs and his joint work on triple cohomology, the classical isomorphism Opext ≅ 𝐻2 that describes group extensions with abelian kernels, can be deduced from the equivalence between such extensions and torsors (in an appropriate sense). The same is known for many other “group-like” algebraic structures, and now we present a purely-categorical version of that equivalence, essentially by showing that all torsors are extensions with abelian kernels in any pointed protomodular category, and by giving a necessary and sufficient condition for the converse.

A general formulation of homotopy limits

The problem of homotopy coherence has occurred recently in two contexts: explicitly in strong shape theory (Edwards-Hastings [ 1 l), Dydak-Segal [lo] etc.) and implicitly in the simplicial localization of Dwyer-Kan (9). In both cases, the authors take into consideration the notion of higher order homotopies (or “homotopy coherences”). One of the first places in which these higher order homotopy coherences have been used is in the study of homotopy limits (in a restricted sense in Bousfield-Kan (61 and Edwards-Hastings [ 11) and more generally Vogt [ 193 and Porter [ 151). These homotopy limits then appear naturally in strong shape theory. Coming from another direction and following an article of Thomason [ 181 which sheds light on possible relations between lax limits, and homotopy limits, Gray [ 131 has introduced a generalization of homotopy limits (in the sense of Bousfield-Kan the precise definition will be given later). This latter definition has two imperfections; it cuts off the coherence at level 2 and it does not allow the generalization of the replacement schemes necessary for the development of the analogues of the Bousfield-Kan spectral sequences. Our own work in shape theory [S] and coherence [4,7,8] has led us to study a fresh definition which remedies these defects. This general definition is not however entirely new as a particular case of it already appears in Segal’s paper [ 161. Like Gray we feel that the best presentations of homotopy limits are made in terms of indexed limits. However for us, the natural context for these indexed limits is that of profunctors (sometimes also called distributors) [14, l] (it is in thzse terms, in fact, that the replacement scheme seems most natural). The first section recalls some necessary facts about Bousfield-Kan homotopy limits and indexed limits. From a careful inspectio.1 of the coherence of a homotopy cone, we introduce in Section 2 a general notion of homotopy limits for a simplicial category. We next show that the replacement scheme 16) holds in this situation and exhibit an indexing for this notion of limit. We give general conditions of existence and study the cases of the two important simplicial categories Cat and Top. In particular we show that lax limits and a construction 01 Segal are particular cases of homotopy limits.

Normal sections and direct product decompositions

Abstract In any finitely complete category, there is an internal notion of normal monomorphism. We give elementary conditions guaranteeing that a normal section s: Y → X of an arrow f: X → Y produces a direct product decomposition of the form X ≃ Y × W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories.

Subtractive Categories and Extended Subtractions

We introduce a notion of an extended operation which should serve as a new tool for the study of categories like Mal’tsev, unital, strongly unital and subtractive categories. However, in the present paper we are only concerned with subtractive categories, and accordingly, most of the time we will deal with extended subtractions, which are particular instances of extended operations. We show that these extended subtractions provide new conceptual characterizations of subtractive categories and moreover, they give an enlarged “algebraic tool” for working in a subtractive category—we demonstrate this by using them to describe the construction of associated abelian objects in regular subtractive categories with finite colimits. Also, the definition and some basic properties of abelian objects in a general subtractive category is given for the first time in the present paper.

Another denormalization theorem for abelian chain complexes

Abstract For each abelian category A and each integer n , it is shown that the category C n A of complexes of length n in A is equivalent to the category n -Grd A of internal n -groupoids in A . Moreover, these denormalization equivalences exchange, up to isomorphism, chain homotopies with what are expected to be n -lax natural transformations. Thus n -groupoids appear to be possible substitutes for n -complexes in the nonabelian context.

A structural aspect of the category of quandles

We show that the category of quandles satisfies a Maltsev property relative to a certain class of split epimorphisms. This structural aspect produces a class of congruences, called puncturing, which have the property to permute with any other congruence.

Polyhedral monadicity of n-groupoids and standardized adjunction

Abstract The problem of n-categorical pasting has received different answers of a geometrico-combinatorial nature. In the case of n-groupoids, this question received here a more synthetical and “geometrico-algebraic” answer, in the form of a monadicity theorem.