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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.
I believe the space of contractions of any contractible space is contractible.
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.)
Don’t you need the higher fillers even just to get contractibility of the space of fillers?
Yes. But I was not stating in which dimensions i was asking for fillers, i.e. all!
I think I just don’t understand what you’re saying. (-:
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.
Ah, I see. Yes, that makes sense.
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