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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.


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Let me just make a quick and not very useful comment: I suspect that references motivated by p-adic (functional) analysis will be very different in content and approach from references motivated by problems over the real/complex field. – Yemon Choi Jul 22 '10 at 0:06
A second not very useful comment: my impression, although I do not know the literature all that well, is that much of the categorical machinery for these quasi-abelian categories (in older work of Raikov they are called semi-abelian categories, this seems not to 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 – Yemon Choi Jul 22 '10 at 0:09
OK, last one for now: if you have seen Prosmans' memoir/article, presumably you are also aware of the long article/exposition of Jean-Pierre Schneiders? – Yemon Choi Jul 22 '10 at 0:11
Do you want to restrict to locally convex TVS, or do you really want all TVS? – Loop Space Jul 22 '10 at 9:07
Thinking more, I'm not sure what you're looking for. Do you want to know how to form limits and colimits in the category of TVS? If so, that's covered in any of the standard texts. Or do you want something more? You seem to imply such at the start, but then your motivation suggests that you are really just interested in the direct construction. – Loop Space Jul 22 '10 at 10:31

Some of the crucial ideas and results on quasi-abelian categories in the context of topological vector spaces are already contained in the work of Lucien Waelbroeck (see here) which predates the work of Schneiders and Prosmans considerably. Waelbroeck's book contains also a lot of examples.

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If I recall correctly, Waelbroeck also refers to some older work of Guy Noel (this is in the context of constructing a kind of abelian envelope of the category of Banach spaces and bounded linear maps, but perhaps the wider category of (LC)TVS is also looked at). – Yemon Choi Aug 8 '10 at 18:58

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