arXiv subjects

Nadja Egner

Publications and source records attributed to Nadja Egner.

3 recordsLinked to original sources

arXiv2024

Double groupoids and $2$-groupoids in regular Mal'tsev categories

We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ of internal $2$-groupoids is a Birkhoff subcategory of the category $ \mathrm{Grpd}^2(\mathscr{C}) $ of double groupoids in a regular Mal'tsev category $\mathscr{C}$ with finite colimits. In particular, when $\mathscr{C}$ is a Mal'tsev variety of universal algebras, the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ is also a Mal'tsev variety, of which we describe the corresponding algebraic theory. When $\mathscr{C}$ is a naturally Mal'tsev category, the reflector from $ \mathrm{Grpd}^2(\mathscr{C}) $ to 2-$ \mathrm{Grpd}(\mathscr{C}) $ has an additional property related to the commutator of equivalence relations. We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ is semi-abelian when $\mathscr{C}$ is semi-abelian, and then provide sufficient conditions for 2-$ \mathrm{Grpd}(\mathscr{C}) $ to be action representable.

math.CT↗

arXiv2024

Galois theory and homology in quasi-abelian functor categories

Given a finite category T, we consider the functor category [T,A], where A can in particular be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fréchet spaces. In this situation, the categories of various internal categorical structures in A, such as the categories of internal n-fold groupoids, are equivalent to functor categories [T,A] for a suitable category T. For a replete full subcategory S of T, we define F to be the full subcategory of [T,A] whose objects are given by the functors G with G(X)=0 for all objects X not in S. We prove that F is a torsion-free Birkhoff subcategory of [T,A]. This allows us to study (higher) central extensions from categorical Galois theory in [T,A] with respect to F and generalized Hopf formulae for homology.

math.CT↗

arXiv2024

A syntactic characterization of weakly Mal'tsev varieties

The notion of a weakly Mal'tsev category, as it was introduced in 2008 by the third author, is a generalization of the classical notion of a Mal'tsev category. It is well-known that a variety of universal algebras is a Mal'tsev category if and only if its theory admits a Mal'tsev term. In the main theorem of this paper, we prove a syntactic characterization of the varieties that are weakly Mal'tsev categories. We apply our result to the variety of distributive lattices which was known to be a weakly Mal'tsev category before. By a result of Z. Janelidze and the third author, a finitely complete category is weakly Mal'tsev if and only if any internal strong reflexive relation is an equivalence relation. In the last part of this paper, we give a syntactic characterization of those varieties in which any regular reflexive relation is an equivalence relation.

math.CT↗