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 $\operatorname{\mathcal{S}}$.
5.5 The $\infty $-Category $\operatorname{\mathcal{QC}}$
Recall that the collection of (small) Kan complexes can be organized into a (large) $\infty $-category $\operatorname{\mathcal{S}}$, given by the homotopy coherent nerve of the simplicial category $\operatorname{Kan}$ (Construction 3.1.6.1). In §5.5.2, we introduce a variant of the $\infty $-category $\operatorname{\mathcal{S}}$ whose objects are pointed Kan complexes $(X,x)$. Here there are (at least) two different ways we might proceed:
Let $\operatorname{Kan}_{\ast }$ denote the category of pointed Kan complexes (Definition 3.2.1.5). Note that $\operatorname{Kan}_{\ast }$ can be identified with the coslice category $\operatorname{Kan}_{\Delta ^{0} / }$, where we regard the standard simplex $\Delta ^{0}$ as an object of the category $\operatorname{Kan}$. This identification determines a simplicial enrichment of the category $\operatorname{Kan}_{\ast }$, and we can obtain an $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } )$ by passing to the homotopy coherent nerve.
If we regard $\Delta ^{0}$ as an object of the $\infty $-category $\operatorname{\mathcal{S}}$, then we can instead form the coslice $\infty $-category $\operatorname{\mathcal{S}}_{ \Delta ^{0} / }$. We will denote this $\infty $-category by $\operatorname{\mathcal{S}}_{\ast }$ and refer to it as the $\infty $-category of pointed spaces (Construction 5.5.2.1).
Beware that the $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}_{\ast } )$ and $\operatorname{\mathcal{S}}_{\ast }$ are not isomorphic as simplicial sets. However, there is a natural comparison functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \hookrightarrow \operatorname{\mathcal{S}}_{\ast }$ which is an equivalence of $\infty $-categories (Proposition 5.5.2.8). This is a special case of a general assertion concerning the compatibility of the homotopy coherent nerve with (co)slice constructions (Theorem 5.5.1.21), which we formulate and prove in §5.5.1.
In §5.5.4, we consider an enlargement of the $\infty $-category $\operatorname{\mathcal{S}}$. Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets and let $\operatorname{\mathbf{QCat}}\subseteq \operatorname{Set_{\Delta }}$ denote the full subcategory spanned by the $\infty $-categories, which we again regard as a simplicial category (see Example 2.4.2.1). The homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathbf{QCat}})$ is an $\infty $-bicategory (Proposition 5.5.4.2), which we will denote by $\operatorname{ \pmb {\mathcal{QC}} }$ and refer to as the $\infty $-bicategory of $\infty $-categories (Construction 5.5.4.1). For many applications, it is convenient to work instead with the underlying $\infty $-category $\operatorname{\mathcal{QC}}= \operatorname{Pith}( \operatorname{ \pmb {\mathcal{QC}} })$, which we study in §5.5.3. Both of these constructions have pointed analogues, which we introduce and compare in §5.5.5.
Structure
- Subsection 5.5.1: Digression: Slicing and the Homotopy Coherent Nerve
- Subsection 5.5.2: The $\infty $-Category of Pointed Spaces
- Subsection 5.5.3: The $\infty $-Category of $\infty $-Categories
- Subsection 5.5.4: The $\infty $-Bicategory of $\infty $-Categories
- Subsection 5.5.5: $\infty $-Categories with a Distinguished Object