# homotopy groups of cubical sets

A cubical set $Box^{op} \to Set$ is a model for a homotopy type, via Grothendieck and Cisinski (here $Box$ is the box category with objects the natural numbers and arrows generated by face and degeneracy maps 'as usual'). A typical example is the singular cubical set of a space, $n \mapsto Hom(I^n,X)$. The homotopy groups of $X$ can be recovered from this cubical set as it satisfies a property analogous to that of Kan complexes (horns have fillers). In general, do the homotopy groups of a cubical set satisfying this 'Kan' condition agree with that of the homotopy type it represents?

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Yes it does. This is completely straightforward, once you know that there is a Quillen equivalence relating cubical sets and simplicial sets: homotopy groups are invariants which can be constructed at the level of the homotopy category, forgetting the models you used to construct it. –  Denis-Charles Cisinski Mar 1 '11 at 21:39
Thanks, Denis-Charles. I think I remember why I asked this question... :) –  David Roberts Mar 2 '11 at 2:31