arXiv subjects
Publications and source records attributed to Dominique Bourn.
In this article, we introduce an extrinsic approach to the notion of semi-direct product, an intrinsic one (namely inside the category Gp of group itself) having been already done elsewhere. This will led us to focus our attention on two algebraic structures (hypersubtraction and hyper-Slominski settings) which will allow us to characterize this extrinsic explicitation.
Starting from the varietal notion of syntactic equivalence relation, we generalized it to a categorical concept; namely Equ-saturating category. We produce various examples and focuse our attention on the protomodular context in which any equivalence relation is then shown to have a centralizer.
We investigate the properties of the Kleisli category KlT of a monad (T,{\lambda},{\mu}) on a category E and in particular the existence of (some kind of) pullbacks. This culminates when the monad is cartesian. In this case, we show that any T-category in E in the sense of A. Burroni coincides with a special kind of internal category in KlT . So, it is the case in particular for T -operads and T -multicategories. More unexpectedly, this, in turn, sheds new lights on internal categories and n-categories.
We investigate the split epimorphisms in the categories of digroups and left skew braces. We show that, unlike the category DiGp of digroups, the category SkB of left skew braces is strongly protomodular. From that, we describe the expected Baer sums of exact sequences of left skew braces with abelian kernel.
Category Theory provides us with a clear notion of what is an internal structure. This will allow us to focus our attention on a certain type of relationship between context and structure.
We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, "product" of ideals of skew braces is often considered. We show that Huq=Smith for left skew braces. Finally, we give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer.
The aim of this work is to point out a strong structural phenomenon hidden behind the existence of normalizers through the investigation of this property in the non-pointed context: given any category E, a certain property of the fibration of points: Pt(E) --> E guarentees the existence of normalizers. This property becomes a characterization of this existence when E is quasi-pointed and protomodular. This property is also showed to be equivalent to a property of the category GrdE of internal groupoids in E which is a kind of opposite, for the monomorphic internal functors, of the comprehensive factorization.
In a regular category $\mathbb E$, the direct image along a regular epimorphism $f$ of a preorder is not a preorder in general. In $Set$, its best preorder approximation is then its cocartesian image above $f$. In a regular category, the existence of such a cocartesian image above $f$ of a preorder $S$ is actually equivalent to the existence of the supremum $R[f]\vee S$ among the preorders. We investigate here some conditions ensuring the existence of these cocartesian images or equivalently of these suprema. They applied to two very dissimilar contexts: any topos $\mathbb E$ with suprema of chains of subobjects or any $n$-permutable regular category.
In (B-Gran, 2004), was given a categorical formulation of the Shifting Lemma which is a characterization of the Congruence Modular Varieties among all the variety of Universal Algebra, introduced in (Gumm, 1983). Starting from a characterization of this Shifting Lemma by a property of the fibers of the fibration of points $¶\EE$ (B, 2005), on the model of what happens for Mal'tsev categories, we shall investigate three directions:\\ 1) a new one: in following the golden thread of abelian split epimorphisms naturally provided by this characterization;\\ 2) a more or less expected one: in measuring the distance between the consequences of the Shifting Lemma in the varietal context and in the much more general categorical one;\\ 3) a quite amazing phenomenon, which I should call Algebraic Crystallography, and which is described in the Introduction.
Mal'tsev categories turned out to be a central concept in categorical algebra. On one hand, the simplicity and the beauty of the notion is revealed through a lot of characterizations of different flavour. Depending on the context, one can define Mal'tsev categories as those for which `any reflexive relation is an equivalence'; `any relation is difunctional'; `the composition of equivalence relations on a same object is commutative'; `each fibre of the fibration of points is unital' or `the forgetful functor from internal groupoids to reflexive graphs is saturated on subobjects'. For a variety of universal algebras, these are also equivalent to the existence in its algebraic theory of a Mal'tsev operation, i.e. a ternary operation $p(x,y,z)$ satisfying the axioms $p(x,x,y)=y$ and $p(x,y,y)=x$. On the other hand, Mal'tsev categories have been shown to be the right context in which to develop the theory of centrality of equivalence relations, Baer sums of extensions, and some homological lemmas such as the denormalized $3 \times 3$ Lemma, whose validity in a regular category is equivalent to the weaker `Goursat property', which has also turned out to be of wide interest.
We investigate what is remaining of the 3x3 lemma and of the denormalized 3x3 lemma, respectively valid in a pointed protomodular and in a Maltsev category, in the context of partial pointed protomodular and partial Maltsev categories, relatively to a class $Σ$ of points. The results apply, among other structures, to monoids, semirings and quandles
We study nilpotency in the context of exact Mal'tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie's functor calculus. We show in particular that the reflection into the subcategory of $n$-nilpotent objects is the universal endofunctor of degree $n$ if and only if every $n$-nilpotent object is $n$-folded. In the special context of a semi-abelian category, an object is $n$-folded precisely when its Higgins commutator of length $n+1$ vanishes.
We introduce the notion of Mal'tsev reflection which allows us to set up a partial notion of Mal'tsevness with respect to a class $\Sigma$ of split epimorphisms stable under pullback and containing the isomorphisms, and we investigate what is remaining of the properties of the global Mal'tsev context. We introduce also the notion of partial protomodularity in the non-pointed context.
We make explicit a larger structural phenomenon hidden behind the existence of normalizers in terms of existence of certain cartesian maps related to the kernel functor.
We analyse some aspects of the notion of algebraic exponentiation introduced by the second author [16] and satisfied by the category of groups. We show how this notion provides a new approach to the categorical-algebraic question of the centralization. We explore, in the category of groups, the unusual universal properties and constructions determined by this notion, and we show how it is the origin of various properties of this category.
We make explicit some conditions on a semi-abelian category D such that, for any abelian group A in D and any object Y in D, the cohomology group homomorphisms with coefficients in A, induced by the inclusion of the abelian objects of D at the level of the slice category D/Y, are actually isomorphisms. These conditions hold in particular when D is the category Gp of groups, and this allows us to give a new insight on the Eilenberg-Mac Lane cohomology of groups. They hold also when D is the category K-Lie of Lie-algebras.