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A few weeks ago, in a discussion on the cafe, Tom Leinster and Mike Shulman (I think Urs and Andrew were involved as well.). were discussing an "algebraic" form of Kan complexes. Is there a page on the nLab about this? If not, is there anywhere I can read up about them?
It's in the comment thread here.
You'll see that Thomas Nikolaus was involved - you can now see his nLab page (linked) for the write-up, and the new post at the cafe (just gone up yesterday or so)
Cool! Thanks for the link!
Um, does this mean that the homotopy hypothesis is proven? Then it should be called the homotopy thesis (theorem? [how about lemma?]), right?
It depends. The homotopy hypothesis is really a condition by which we judge definitions of n- or oo-groupoids. But one could say that it was proven when we knew Kan complexes (of the ordinary sort) modeled all spaces up to weak homotopy type, if you take the same position as some where oo-groupoids are Kan complexes.
Personally I would call it to be 'done' when we show globular (eg batanin) oo-groupoids model all (weak) homotopy types, at least from Grothendieck's point of view, as this was the original formulation in AG's letter to Breen in 1975. Then the challenge remains to show that all definitions of oo-groupoids are equivalent, say by showing that the (oo,1)-categories of such are equivalent .
Now that we have an algebraic form, shouldn't it be a straight shot to the finish? I mean, the proof shows that it's algebraic (in the monadic sense), that seems like enough to work with to bridge the gap from the other direction, or am I vastly oversimplifying things. To be honest, I know very little about the algebraic side of higher category theory. The only thing I know is that operads are somehow a unifying "device" in the whole affair.
Harry, it's harder than that! But algebraic Kan complexes do seem like they may be helpful in making a connection to Batanin's oo-groupoids. Several of us are thinking in that direction right now.
I don't think there is any "the" homotopy hypothesis. There is only "the homotopy hypothesis for X definition of n-category." The homotopy hypothesis for the definition "an oo-groupoid is a Kan complex" is proven. By Thomas' result, the homotopy hypothesis for the definition "an oo-groupoid is an algebraic Kan complex" is now also proven.
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