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I have just added a link to the notes that I prepared for the Lisbon meeting on my personal page. I would love to have some feedback, and in particular suggestions for incorporating some more of this in the nLab. The new material also forms part of the extended version of the Menagerie (which is now topping 800 pages.)
Which part of that document did you deliver in Lisbon?
Is there anything resembling a cobordism hypothesis for your $HCobord(n,B)$?
I gave very little but decided to make it available so that people had some notes giving references at least to the discrete cases. (I only had 4 x 45 minutes plus a final 35 minutes) so I gave an intro to n-types (Postnikov towers etc.), then a brief trip through simplicial theory then crossed modules and crossed n-cubes, finishing up with HQFTs.In the 35 mins I tried to connect things up with ideas from other talks.
I briefly talked about cobordism hypotheses for B-Hqfts with Chris Schommer-Pries and others. There seems to be a good chance that much of the methodology would go through, but first we need EHQFTs etc. Some people (including Urs I think) mentioned TQFTs for manifolds with structure and that is in very much the same area but the HQFTs only handle topological structure, e.g. spin structures, and so on.
There was a lot of discussion going on so I hope I have conveyed some of the threads.
In case anyone comments: the version on the link of the notes is constructed from the Menagerie by including selected chapters but there is no attempt to fix cross references (hence some ??), the later material on HQFTs will be an edited (and hopefully improved, updated, expanded, etc.) version of published stuff, but as yet the editing is ’in progress’ so there is duplication due to reorganisation, etc. so I know (some of) its imperfections. I would appreciate suggestions for further topics. (I would like to include the crossed $G$-modular category stuff from Turaev and to explore how that might extend to higher dimensions, but need to get the early stuff sorted out first. Also a discussion in some early chapters of the cobordism hypothesis would be good as then David C’s question could be examined. I think that the methods of proof given by Lurie should be fairly easily generalised, and Thomas’s talk in Lisbon suggested some of the points that needed handling.)
I briefly talked about cobordism hypotheses for B-Hqfts with Chris Schommer-Pries and others. There seems to be a good chance that much of the methodology would go through, but first we need EHQFTs etc. Some people (including Urs I think) mentioned TQFTs for manifolds with structure and that is in very much the same area but the HQFTs only handle topological structure, e.g. spin structures, and so on.
I have added to cobordism hypothesis remarks and references on the following:
In On the Classification of TFTs there is the definition of $Bord_n^{(X,\xi)}$ of cobordisms "with $(X, \xi)$-structure". If one takes here $X = B SO(n) \times Y$ for any topological space $Y$, the result is close to being cobordisms in the sense of HQFT over $Y$.
In a recent PhD thesis by David Ayala he considers cobordisms categories (unextended) of cobordisms equipped with geometric structure given by maps into, more or less, an smooth $\infty$-groupoid $X$.
Thanks. In fact I think I had looked at that and was surprised that Ayala had not seen the connection with Turaev’s work.
was surprised that Ayala had not seen the connection with Turaev’s work.
Well, what Ayala considers goes in another direction, representations of his cobordisms are really not topological QFTs anymore.
But I think Jacob Lurie would have had a chance to mention Turaev’s work around where he discusses cobordisms with $(X,\xi)$-structure. I think this is a generalization of HQFTs and provides the extended case. Maybe there is some technical fine-tuning necessary concerning the emphasis of the base point in Turaev’s description, I haven’t checked that (but it should not matter, should it?)
@arsmath - see post #4 for the disclaimer. If you email Tim he will most likely send you a full copy.
Yep! If you are looking at the Lisbon notes version or for that matter the ’short’ version on the nLab, and want to get a longer version, just ask me. It is work in progress and it becomes tedious to do a fresh delete and upload of the new version every few days.
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