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Atrium > Mathematics, Physics & Philosophy: polysets and T-complexes

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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2011

    I just started having a look at Jones’ thesis on “Poly T-complexes” cited at T-complex, in the hope that the general shapes he considers might be useful in “combinatorializing” the notion of disc-like n-category. As previous related work, he cites a University of Warwick MPhil thesis of S. Hintze called “Polysets, 2-sets and semi-cubical sets”, and his description of Hintze’s “polysets” sounds vaguely similar to what Morrison and Walker call “cone-product sets”. Does anyone know where I can obtain a copy of Hintze’s work?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2011

    I don’t, but BTW: did you find the answer to the question on blob-categories, that you asked recently?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2011

    Yes, Kevin emailed me privately and we worked it out. The definition given in their paper at the time required a little bit of fixing, but after that I was able to follow his argument for the unit laws.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeJun 20th 2011

    I have suggested to Ronnie that he chip in here. He probably still has Hintze’s thesis.

    • CommentRowNumber5.
    • CommentAuthorronnie
    • CommentTimeJun 20th 2011
    It is possible that Brian Sanderson (now retired) at Warwick can get hold of a copy. It may be possible to find the copy I once had, but it is really the responsibility of Warwick to make it available. As far as I remember, the method was a variation of Gabriel-Zisman. I can't remember what she did on poly-sets. Another generalisation of cubical and simplicial is

    Gugenheim, V. K. A.~M.
    {On supercomplexes}.
    {Trans. Amer. Math. Soc.} \textbf{85} (1957) 35--51.

    but it does not do realisations, I think.

    For further work on cubical realisations, see
    Maltsiniotis, G.
    {La cat\'egorie cubique avec connexions est une cat\'egorie test stricte}.
    {Homology, Homotopy Appl.} \textbf{11}~(2) (2009) 309--326.

    and the references there, particularly Cisinski.

    Delighted some are looking at David Jones' thesis!

    Ronnie
    • CommentRowNumber6.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 20th 2011
    A former student of mine, Roberto Ruiz, did his thesis on singular vs realization
    in great if not maximal generality. Will see if I can find access to his thesis.
    • CommentRowNumber7.
    • CommentAuthorronnie
    • CommentTimeJun 20th 2011
    Another comment is that David Jones' work is aimed at the groupoid theory, and so does not at all touch the question of the use of variants of Poly-sets for higher order category rather than groupoid theory. Presumably it needs some input from the ideas of Verity and of Steiner on complicial sets; that is, if there is some kind of ordering, then the notion of what horns can have fillers need to be refined, and indeed there needs to be an analysis what kind of marking is needed, and what kinds of such marked cells are allowed.

    Ronnie
    • CommentRowNumber8.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 23rd 2011
    a reprint from
    Revista de la Academia Colombiana de Ciencias
    Vol. 28 (106) (2004), 100-121
    OVERVIEW ON MODELS IN HOMOTOPICAL
    ALGEBRA
    by
    Roberto Ruiz S.

    This is an overview of part of his thesis.
    Here models refers to simplices or cubes or fairly general
    suitable for singular/realization adjointness.

    I can supply a copy either to individuals or for posting here or on the blog or...

    The thesis is available via mathscinet
    just search for Ruiz, Roberto



    MR0574099 (58 #28140) Ruiz, Roberto The closure of a model category. Rev. Colombiana Mat. 11 (1977), no. 1-4, 19–50, 18F99 (55D99)
    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJun 23rd 2011

    Huh, somehow a bunch of posts here didn’t show up in my RSS. Thanks for all the references! Although at the moment I’m mostly just interested in the definition of “polysets”, to compare it with “cone-product sets”.

    • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeMar 6th 2012
    • (edited Mar 6th 2012)

    I noted that singular complex was grey on an entry, so did some redirects. The point was that there was no really linking into thin from the simplicial set entry. There was an actual link but the wording seemed not to be quite optimal. I have tried to fix them so give these entries a glance, adjust etc.

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