Examples and non-examples of integral categories and the admissible intersection property

Abstract

Integral categories form a sub-class of pre-abelian categories whose systematic study was initiated by Rump in 2001. In the first part of this article we determine whether several categories of topological and bornological vector spaces are integral. Moreover, we establish that the class of integral categories is not contained in the class of quasi-abelian categories. In the last part of the article we show that a category is quasi-abelian if and only if it has admissible intersections, in the sense considered recently by Brüstle, Hassoun and Tattar. This exhibits that a rich class of non-abelian categories having this property arises naturally in functional analysis.A pre-abelian category is an additive category in which every morphism has a kernel and a cokernel. Within the class of pre-abelian categories, one defines the following notions. Semi-abelian categories are the ones in which the canonical morphism between the coimage and image is always both monic and epic, but not necessarily an isomorphism as one would expect in an abelian category. Quasi-abelian categories are the ones in which