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    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeMar 9th 2011

    Beren had corrected a typo in geometric definition of higher categories so I looked again at that entry. Two points / questions came to mind that may be worth airing here.

    (i) In a geometric model the composition maps ’composable horns’ to fillers in some way and is a relation. There is a contractibe space of composites. Has anyone thought of putting a probability measure on that space of fillers? What would be reasonable axioms to require? I encountered this sort of situation when looking at nerves of covers of Chu spaces, as I would have loved to replace the values used (in the usual case {0,1}\{0,1\} by something like [0,1][0,1] with combination using some probability or t-norm. This leads to simplicial complexes in which simplicies have a probability of existing, rather than either being there or not (if you see what I mean). Has anyone seen anything like this in the literature? (It is disantly related to some of Tom’s musing in the Café.)

    (ii) Am I right to say that the space of composites is not only contractible but its space of contractions is contractible and then …. . :-) i.e. the current definition sells short the structure of a Kan complex or quasi-category by some distance.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMar 9th 2011

    I believe the space of contractions of any contractible space is contractible.

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeMar 9th 2011

    I suspect that is true, but my point is that the Kan complex definition makes the implications of that important, as it says not only that a filler exists but that any two fillers are related by a higher filler etc. In algorithmic version of this which are somehow half algebraic half geometric, there is a sense of … suppose you have a machine (non-deterministic) that given a box gives a filler then …. (I rather like the idea of a non-deterministic machine giving the filler etc, but do not immediately see what it would give one.)

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMar 9th 2011

    Don’t you need the higher fillers even just to get contractibility of the space of fillers?

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeMar 9th 2011

    Yes. But I was not stating in which dimensions i was asking for fillers, i.e. all!

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMar 10th 2011

    I think I just don’t understand what you’re saying. (-:

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeMar 10th 2011

    Oh dear, I must be writing rubbish! It was meant to be an obvious comment and I think I have made it more complicated in my explanation than I intended. It is not intended to be particularly deep, just a remark. Here is perhaps a better attempt.

    At any level a (n,k)-horn will have a filler, and any two such fillers are linked in a (n+1, something)-horn so are equivalent by a filler in the next dimension.The precise ’something’ will depend on the way in which that horn is constructed. So one has a very high level of coherence that if one has algorithms for filling can be made as deterministic /algebraic as the initial filling algorithm (e.g. in the underlying simplicial set of a simplicial group, GG, and also in W¯G\overline{W}G. So not only is the space of fillers contractible it is potentially algorithmically contractible in some sense.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMar 10th 2011

    Ah, I see. Yes, that makes sense.