$\newcommand\Box{\mathrm{Box}}\newcommand\Set{\mathrm{Set}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}$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?



Homotopy groups of cubical sets
I think a reference for this would be [Theorem 3.24](https://arxiv.org/pdf/2202.03511.pdf#page=26) of



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> 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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