We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric realization functor. We also provide purely combinatorial proofs of several classical theorems, including: product preservation, commutativity of higher homotopy groups, the long exact sequence of a fibration, and Whitehead’s theorem.

This is a companion paper to our “Cubical setting for discrete homotopy theory, revisited” in which we apply these results to study the homotopy theory of simple graphs.

中文翻译：

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立方集的同伦群

我们定义并研究立方集的同伦群。为此，我们给出了立方集同伦群的四个定义，证明它们是等价的，并进一步通过几何实现函子证明它们与它们的拓扑类似物一致。我们还提供几个经典定理的纯组合证明，包括：乘积保存、高同伦群的交换性、纤维化的长精确序列和怀特海定理。

这是我们的“离散同伦理论的立方设置，重新审视”的姊妹篇，其中我们应用这些结果来研究简单图的同伦理论。