$\newcommand\Box{\mathrm{Box}}\newcommand\Set{\mathrm{Set}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}$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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asked Feb 25, 2010 at 2:10
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$\begingroup$ 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. $\endgroup$– D.-C. CisinskiCommented Mar 1, 2011 at 21:39
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$\begingroup$ Thanks, Denis-Charles. I think I remember why I asked this question... :) $\endgroup$– David Roberts ♦Commented Mar 2, 2011 at 2:31
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2$\begingroup$ In fact Dan Kan's thesis and first paper were cubical, but at Princeton it was felt that because cubical groups were not Kan and the realisation of the cartesian product of cubical sets had the wrong homotopy type, meant that the cubical theory had to be abandoned. These deficiencies have been remedied by cubical sets with connections, which are used extensively in our big book on Nonabelian Algebraic Topology, because of other advantages of cubical methods. This idea has not been taken up in current $\infty$-category theory. $\endgroup$– Ronnie BrownCommented Oct 15, 2015 at 14:23
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$\begingroup$ @RonnieBrown I should point out, and I'm sure you know this, that cubical methods are being taken up by the UF/HoTT community (Thierry Coquand is leading this effort, I think) for the purposes of finding good models of UF/HoTT. One can see higher category theory natively in HoTT, in that is captures (higher) categories of 'sheaves of homotopy types'. $\endgroup$– David Roberts ♦Commented Oct 15, 2015 at 20:55
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$\begingroup$ @David: I was aware of this and gave a talk at Paris in 2014 at the invitation of Thierry (see my preprint page for that and other expositions.) The more subtle point in my and colleagues' work is the use of structured spaces, i.e. "topological data", rather than just spaces. Can this be mirrored more widely? $\endgroup$– Ronnie BrownCommented Jul 21, 2016 at 10:48
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I think a reference for this would be Theorem 3.24 of
Homotopy groups of cubical sets, Daniel Carranza, Chris Kapulkin, 2022. arXiv:2202.03511, https://doi.org/10.48550/arxiv.2202.03511
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1$\begingroup$ Actually, I think Theorem 3.25 might be even more relevant, in combination with Corollary 2.24, which ensures the the singular/geometric realisation adjunction is a Quillen equivalence with the standard Kan–Quillen model structure on $\mathbf{Top}$. $\endgroup$– David Roberts ♦Commented Nov 27, 2022 at 6:03