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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2018

    made explicit the conclusion that forming Lie groupoid convolution algebras is a (2,1)-functor

    C *(−):DiffStack prop⟶AAAC *Alg bim op C^\ast(-) \;\colon\; DiffStack^{prop} \overset{\phantom{AAA}}{\longrightarrow} C^\ast Alg^{op}_{bim}

    here

    diff, v62, current

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeOct 29th 2021

    Fixed some links

    diff, v65, current

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeOct 29th 2021

    Fixed some more accented links

    diff, v65, current

    • CommentRowNumber4.
    • CommentAuthorJosh
    • CommentTimeMar 30th 2023
    • (edited Mar 30th 2023)
    Is there an example of a higher groupoid convolution algebras that can be discussed, or can more details be given? I have an example in mind that I'd like to understand...or is this only for double groupoids?
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2023

    I am not aware of further developments since I made that note in April 2013 (revision 46). But it could well be that there was development which I didn’t hear of.

    What’s the example you have in mind?

    • CommentRowNumber6.
    • CommentAuthorJosh
    • CommentTimeMar 31st 2023
    • (edited Apr 1st 2023)
    I haven't been able to find anything either, but perhaps the functor which sends groupoids to algebras sends higher groupoids to higher (a-infinity?) algebras? The examples I have in mind come from Poisson manifolds — we can specialize to symplectic manifolds. The fundamental 2-groupoid of such a manifold comes with a 2-cocycle and one can form the twisted convolution "algebra", which is related to Kontsevich's star product and the Poisson sigma model via formula (3.39) in https://arxiv.org/abs/math/0507223, as I explain in section 4.4 and 5 in https://arxiv.org/pdf/2303.05494.pdf...but basically you make the most obvious construction of something that's like an algebra out of the functions on the 1-arrows of the 2 groupoid that reproduces the convolution algebra on 1-groupoids... However, it's not clear that the product is associative (there is still a way of obtaining an associative algebra, but there may be something better that one can do). I wonder if there are simple examples of finite 2-groupoids (modeled as kan-complexes) which aren't strict that one can play with.

1 to 6 of 6

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