It's known that category of topological vector spaces is not abelian but quasi-abelian or exact category. I am looking for the references playing with this category(category theory). All the related references are welcome which means

The abstract work on quasi abelian category is ok, although I am more interested in some concrete examples, some special topological vector spaces.

I realized that it seems that p-adic analysis and p-adic representation theory has something to do with this kind of stuff, however, I do not know anything about this area.

I am not sure whether Grothendieck's work on topological vector spaces writing about something on this category. Somebody told me that Grothendieck formulated the algebraic inductive limits and many other categorical constructions in his book. I have not checked out this book to see.

My motivation for asking this question is that I am considering unitary representation of compact group which are Hilbert spaces. I want to consider category of unitary representations of this group and formulate some categorical construction, such as limits, colimits and so on so forth which might be helpful to study representation theory.

I noted that there are some work by Fabienne Prosmans on derived category and functional analysis.

Thanks!

notto be usefully related to Borceux-Bourn et al's definition of semi-abelian) is somehow "folklore" or "done on an ad hoc basis". It is Stuff Every Functional Analyst Picks Up, but I don't know where the categorical aspects are explicitly written down $\endgroup$locally convexTVS, or do you really wantallTVS? $\endgroup$1more comment