Effective descent morphisms in star-regular categories

Abstract

In this article a sufficient condition on a star-regular category is introduced guaranteeing that regular epimorphisms are effective descent morphisms. This condition is satisfied by any category with a good theory of ideals (thus, in particular, by any ideal determined category), by any almost abelian category (for instance, by the categories of torsion abelian groups, torsion-free abelian groups, normed vector spaces, Banach spaces, locally compact abelian groups, etc.) and by any category of topological Mal'tsev algebras (in particular, by the category of topological groups).

Star-regularity and regular completions

“…From the syntactic characterisation of 0-regular varieties obtained in [8], the following property of such varieties can be easily deduced: the congruence generated by the 0-class of a reflexive homomorphic relation always contains the reflexive relation as a subrelation. In [13] it was shown that a similar fact holds more generally in any star-regular category. In this paper we refine and further generalise this result -see Theorem 2.2 below.…”

“…So, since a regular epimorphism is always a coequalizer of its kernel pair, and since any kernel pair in a regular category has a coequalizer, star-regularity states precisely that kernel pairs satisfy ( * π 0 ). In [13] it was shown that in a star-regular category every reflexive relation satisfies ( * π 0 ). In fact, we have: Theorem 2.2.…”

“…A weak kernel pair is, in particular, a reflexive graph, so (a)⇒(b) is obvious. For (b)⇒(a) we adapt the argument given in [13] to the present context: consider the display…”

Some remarks on pullbacks in Gumm categories

“…For instance, this is the case for any category of topolo-gical Mal'tsev algebras [19], then in particular for the category of topological groups. Also any almost abelian category in the sense of [21] is almost exact (see [12]). Another example of almost exact category is provided by the category of regular epimorphisms in an exact Goursat category [16].…”

Fundamental group functors in descent-exact homological categories

“…Therefore Gp(Top) is neither a semi-abelian category nor an almost abelian category. Nevertheless, it is known that every topological semi-abelian variety is (cocomplete and) homological [4, Theorem 50] and descent-exact [29,Section 4.5]. Note that the categories of models of semi-abelian theories in the category of compact Hausdorff spaces are themselves semi-abelian categories [4,Theorem 50].…”