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On the nLab page on n-fold categories one finds the following statement:
Even though an n-fold category is a strict version of an n-category in that all n composition operations are strictly unital and associative and strictly commute with each other, still n-fold groupoids model all homotopy n-types. See homotopy hypothesis.
But at homotopy hypothesis there is no mention of n-fold groupoids, and one is pointed instead to n-groupoids. So I figure that in some sense n-groupoids and n-fold groupoids are equivalent notions. But which is a precise statement?
Hi Domenico,
that page should be further clarified (but I don’t have time for it now).
Strict n-fold groupoids are equivalent to strict n-groupoids if the horizontal, vertical etc. 1-groupoids are all the same, and if all the expected cubical identity cells (thin cells) are present. Otherwise strict n-fold groupoids are more general, and it is this generality that makes them model more homotopy types than strict n-groupoids.
Specifically this result is traditionally formulated for connected homotopy types and “cat-n-groups”. That page has a few more relevant citations.
But let me ask: what is it you are after? Depending on what it is, we might have better hints for how to proceed.
But at homotopy hypothesis there is no mention of n-fold groupoids,
… there is now! The point is that cat-n-groups are $n$-fold categories internal to the category of groups, hence are $n+1$-fold groupoids. I have added some links and a bit more to various pages, but really this could do with more.
Hi Tim,
thanks!
But let me ask: what is it you are after? Depending on what it is, we might have better hints for how to proceed.
I’m interested in the realization of a $n$-homotopy type as an $n$-fold groupoid. In particular, given a nice topological space $X$, I’d like to have a $n$-fold groupoid presentation of its Poincare’ $n$-groupoid $\Pi_n(X)$.
Domenico, In that case I’d go for either the cubical set of singular cubes, regarded in the model structure on cubical sets, or if making the $n$-fold-ness is crucial for your purpose, then I’d consider the n-fold complete Segal space of singular $n$-cubes.
An alternative is to work with a simplicially enriched setting. In that case (and I will assume that the homotopy type is connected so as to cheat and talk of a simplicial group), there are explicit (and very simple) formulae for the corresponding crossed n-cube, and simple relations between the models for different n. The advantage with this is that n-simplicial groups collapse nicely back to simplicial groups (using the codiagonal). That aspect is very nicely put in the parallel theory developed by Bullejos, Cegarra, and Duskin. You can fold up the n-simplicial group into a simplicial group or take its classifying space. (A lot is done on this by Cegarra and various collaborators.)
The idea of a n-fold presentation of a Poincaré n-groupoid may correspond to the collapsing process that I mention above.
I thought I had put the abstract on this n-forum but have not yet found it!
You had put it here: http://nforum.ncatlab.org/discussion/177/homotopy-ntype/?Focus=53719#Comment_53719
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