Kerodon

Warning 5.5.0.1. The constructions of this section depend on a choice of dichotomy between “small” and “large” mathematical objects, and we implicitly assume that the categories $\operatorname{Set_{\Delta }}\supseteq \operatorname{\mathbf{QCat}}\supseteq \operatorname{Kan}$ consist only of small simplicial sets. In particular, the objects of $\operatorname{\mathcal{S}}$ are small Kan complexes, and the objects of $\operatorname{\mathcal{QC}}$ are small $\infty $-categories. By contrast, the $\infty $-categories $\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{QC}}$ are not themselves small. In particular, one cannot regard $\operatorname{\mathcal{QC}}$ as an object of itself, or the Kan complex $\operatorname{\mathcal{S}}^{\simeq }$ as an object of $\operatorname{\mathcal{S}}$.