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5.5 The $\infty $-Categories $\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{QC}}$

Let $\operatorname{Kan}$ denote the category of Kan complexes and let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote its homotopy category (Construction 3.1.4.10). There is an evident forgetful functor $U: \operatorname{Kan}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$, which carries each Kan complex $X$ to itself and each morphism of Kan complexes $f: X \rightarrow Y$ to its homotopy class $[f] \in \pi _0(\operatorname{Fun}(X,Y))$. Broadly speaking, homotopy theory is concerned with questions about Kan complexes which are invariant under homotopy equivalence. Since a morphism of Kan complexes $f$ is a homotopy equivalence if and only if its homotopy class $[f]$ is an isomorphism, it is tempting to characterize homotopy theory as the study of the category $\mathrm{h} \mathit{\operatorname{Kan}}$. Beware that this characterization is somewhat misleading: many questions belonging to the purview of homotopy theory cannot be formulated at the level of the homotopy category. For example, suppose we are given a commutative diagram of Kan complexes $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d] & X \ar [d] \\ Y' \ar [r] & Y. } \]

One can then ask if $\sigma $ is a homotopy pullback square (Definition 3.4.1.1). Though the answer to this question depends only on the homotopy type of the diagram $\sigma $ (Corollary 3.4.1.10), it does not depend only on the associated diagram $U(\sigma )$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Example 3.4.1.11).

Roughly speaking, the problem is that passage from the category of Kan complexes $\operatorname{Kan}$ to its homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ destroys too much information. To remedy the situation, it is convenient to consider a refinement of the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Note that $\operatorname{Kan}$ has the structure of a simplicial category (see Example 2.4.2.1). In §5.5.1, we show that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$ is an $\infty $-category (Proposition 5.5.1.2), which we will denote by $\operatorname{\mathcal{S}}$ and refer to as the $\infty $-category of spaces (Construction 5.5.1.1). After passing to nerves, the forgetful functor $U: \operatorname{Kan}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ factors as a composition

\[ \operatorname{N}_{\bullet }(\operatorname{Kan}) \xhookrightarrow {U'} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}\xrightarrow {U''} \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{Kan}}) \]

with the following features:

  • The functor $U'$ is a monomorphism of simplicial sets which is bijective on vertices and edges. In particular, we can identify objects of the $\infty $-category $\operatorname{\mathcal{S}}$ with Kan complexes, and morphisms in the $\infty $-category $\operatorname{\mathcal{S}}$ with morphisms of Kan complexes (Remark 5.5.1.3).

  • The functor $U''$ exhibits $\mathrm{h} \mathit{\operatorname{Kan}}$ as a homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}$ (Remark 5.5.1.4). In particular, a map of Kan complexes $f: X \rightarrow Y$ is a homotopy equivalence if and only if it is an isomorphism when regarded as a morphism of the $\infty $-category $\operatorname{\mathcal{S}}$ (Remark 5.5.1.5).

In §5.5.3, 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 3.2.1.1). Note that $\operatorname{Kan}_{\ast }$ can be identified 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 5.5.3.1).

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 5.5.3.5). This is a special case of a general assertion concerning the compatibility of the homotopy coherent nerve with (co)slice constructions (Theorem 5.5.2.21), which we formulate and prove in §5.5.2.

In §5.5.5, 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 $(\infty ,2)$-category (Proposition 5.5.5.2), which we will denote by $\operatorname{ \pmb {\mathcal{QC}} }$ and refer to as the $(\infty ,2)$-category of $\infty $-categories (Construction 5.5.5.1). 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 §5.5.4. Both of these constructions have pointed analogues, which we introduce and compare in §5.5.6.

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 $\operatorname{\mathcal{S}}$. We will return to this point in §.

Structure

  • Subsection 5.5.1: The $\infty $-Category of Spaces
  • Subsection 5.5.2: Digression: Slicing and the Homotopy Coherent Nerve
  • Subsection 5.5.3: The $\infty $-Category of Pointed Spaces
  • Subsection 5.5.4: The $\infty $-Category of $\infty $-Categories
  • Subsection 5.5.5: The $(\infty ,2)$-Category of $\infty $-Categories
  • Subsection 5.5.6: $\infty $-Categories with a Distinguished Object