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nLab > Latest Changes: stereotype space

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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 9th 2013

    Someone anonymously started stereotype space. How well-used is this notion?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 9th 2013

    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).

    • CommentRowNumber3.
    • CommentAuthorYemon Choi
    • CommentTimeNov 10th 2013

    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

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 10th 2013

    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.)

    • CommentRowNumber5.
    • CommentAuthorYemon Choi
    • CommentTimeNov 10th 2013

    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.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 10th 2013

    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?

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeNov 10th 2013

    I thought something with this property was called “reflexive”.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 10th 2013

    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.

    • CommentRowNumber9.
    • CommentAuthortrent
    • CommentTimeApr 10th 2016

    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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