  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, and that
there exist semi-abelian categories that are neither integral nor
quasi-abelian. 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 Br{\"u}stle, Hassoun and Tattar. This exhibits that a
rich class of non-abelian categories having this property arises naturally in
functional analysis.
