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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 $\{0,1\}$ by something like $[0,1]$ 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, $G$, and also in $\overline{W}G$. 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.