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nLab > Latest Changes: Feynman category

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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeMay 28th 2018
    • (edited May 28th 2018)

    The definition and the annotated bibliography are given for Feynman category.

    v1, current

    I wonder how useful this could be in related to elucidate the cohomological and motivic quantization via correspondences (Kan extensions in the setup of Feynman categories can help getting the pushforwards, Connes-Kreimer Hopf algebra, Feynman transform (which in some cases gives coefficients in the formal development of the Feynman integal, basically being partition functions, hence connection to graphs).

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 30th 2019

    Not much the wiser as to what they’re for, but I added the recent

    • Ralph M. Kaufmann, Feynman categories and Representation Theory, (arXiv:1911.10169)

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 30th 2019

    I made the definition tally with

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2019
    • (edited Dec 1st 2019)

    Not much the wiser as to what they’re for,

    Feynman categories are just another way of saying “colored operad”. This was proven in Batanin-Kock-Weber 15. I have now highlighted this a bit more in the entry, both in an brief Idea-section and in a Properties section (both telegraphic for the time being, just meant to point to that reference)

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 1st 2019

    I see. So rather an unnecessary name.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2019
    • (edited Dec 1st 2019)

    An independent proof of the result is also claimed in section A.1 of

    (This appeared as a preprint in the same month, just a little earlier. It cites Batanin et al. as “in preparation”. But then Caviglia’s article seems not to have been published(?))

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2019

    So rather an unnecessary name.

    Possibly. I suppose it was motivated from the previously established term Feynman transform.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeDec 1st 2019

    … and “colored operad” is just another way of saying “symmetric multicategory”.

    Usually when a concept has more than one name, we only have one page about it. Should we merge Feynman category with operad or symmetric multicategory?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2019
    • (edited Dec 1st 2019)

    I’d think the explicit definition of “Feynman category” seems sufficiently different from that of operad/multicategory to not be the same concept under a different name. The proof of their bi-equivalence seems to be rather non-trivial. (?) In this case it seems better to me to keep separate entries, albeit with clear cross-linking.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 1st 2019

    An example of where we have different pages for equivalent concepts is clone and Lawvere theory. The difference is in the presentation.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeDec 2nd 2019

    Ah, I see; the 2-categories are only biequivalent. It makes sense to keep the pages separate then.

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