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Someone anonymously started stereotype space. How well-used is this notion?
It’s the first I’d heard of it (which might not mean much). But I have to say that it does look very interesting, and I’m glad the Anonymous Coward wrote something (even if it’s just a copy and paste from Wikipedia).
Promoted by Sergei Akbarov in recent years, IIRC. Am a bit busy right now but someone could probably dig up the title, if not the content, of his relevant articles through MRLookup
I’ve added more material to stereotype space. A main result (hinted at but not quite said in the Wikipedia article) is that stereotype spaces form a star-autonomous category. (It’s amazing to me how people keep rediscovering this concept in disparate areas, without realizing that categorists have had the general notion for 35 years.)
Wasn’t Michael Barr motivated by the existing theorems in functional analysis, in particular the Mackey topology on the dual of a TVS? I haven’t read his paper(s) properly and when I did it was some time ago.
Indeed, that is my understanding as well, Yemon. So clearly I’m not trying to lord the category theorists over the functional analysts; I was suggesting that perhaps some of the functional analysts aren’t aware of the abstract niches that have been developed starting from those theorems. Or is that unfair?
I thought something with this property was called “reflexive”.
Mike, yes I guess that is so, except that I’m not sure if “reflexive” is reserved for a particular sense of “dual”. So I’ll hold off adding that in.
just came across akbarov’s work via his extremely interesting recent mathoverflow answer:
Recently a new application of functional analysis in geometry appeared, the study of envelopes of topological algebras. It allows to look at “big” geometric disciplines – complex geometry, differential geometry, topology – from the point of view of category theory, so that these disciplines become “purely categorical constructions”. This can be considered as a developement of Klein’s Erlangen program.
According to this view, different geometric disciplines are just pictures that appear in the imagination of an outlooker after applying different “observation tools” for studying a given category of topological algebras. Formally, this “projection of functional analysis to geometry” is established by a categorical construction, called envelope (and there are many different envelopes that give different geometries as disciplines).
This activity allows to build different generalizations of Pontryagin duality to classes of non-commutative groups (including some quantum groups).
checked the nlab/nforum and this is the only page that mentions him. is his work well known at all in 2016? fairly obscure?
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