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8.2.4 Strict Models for Hom-Functors

Let $\operatorname{\mathcal{E}}$ be a (locally small) $\infty $-category. Proposition guarantees the existence of a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$, which is well-defined up to isomorphism. Our goal in this section is to give an explicit construction of a $\operatorname{Hom}$-functor in the special case where $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ arises as the homotopy coherent nerve of a (locally Kan) simplicial category $\operatorname{\mathcal{C}}$.

Construction Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. Then the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ determines a simplicial functor $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$. Passing to homotopy coherent nerves, we obtain a functor of $\infty $-categories

\[ \mathscr {H}_{\operatorname{\mathcal{C}}}: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}. \]

Proposition Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. Then the functor $\mathscr {H}_{\operatorname{\mathcal{C}}}$ of Construction is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$.

Remark Let $\operatorname{\mathcal{C}}$ be an ordinary category, which we identify with the corresponding constant simplicial category (see Example In this case, Proposition reduces to Example

Remark By combining Proposition with the rectification results of ยง, we can give an explicit construction of a $\operatorname{Hom}$-functor for an arbitrary (small) $\infty $-category $\operatorname{\mathcal{E}}$. Let $\operatorname{Path}[\operatorname{\mathcal{E}}]_{\bullet }$ denote the simplicial path category of $\operatorname{\mathcal{E}}$ (Definition and let $\operatorname{\mathcal{C}}$ be the locally Kan simplicial having the same objects, with morphism spaces given by $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } = \operatorname{Ex}^{\infty }( \operatorname{Hom}_{ \operatorname{Path}[\operatorname{\mathcal{E}}]}(X,Y)_{\bullet } )$ (see Example ). It follows from Proposition that the tautological map $\operatorname{Path}[ \operatorname{\mathcal{E}}]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ is a weak equivalence of simplicial categories (in the sense of Definition, and therefore corresponds to an equivalence of $\infty $-categories $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (Theorem ). Using Proposition, we deduce that the composition

\[ \operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}\xrightarrow { F^{\operatorname{op}} \times F} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \xrightarrow { \mathscr {H}_{\operatorname{\mathcal{C}}} } \operatorname{\mathcal{S}} \]

is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{E}}$, given on objects by $(X,Y) \mapsto \operatorname{Ex}^{\infty }( \operatorname{Hom}_{ \operatorname{Path}[\operatorname{\mathcal{E}}] }(X,Y) )$.

Beware that, although this construction is completely explicit in principle, it is hard to use in practice (since the operations $\operatorname{\mathcal{E}}\mapsto \operatorname{Path}[\operatorname{\mathcal{E}}]_{\bullet }$ and $S \mapsto \operatorname{Ex}^{\infty }(S)$ are both difficult to control).

Proposition asserts that the functor $\mathscr {H}_{\operatorname{\mathcal{C}}}$ is a covariant transport representation for the left fibration $\operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (Remark We will prove this by constructing a categorical pullback square of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) ) \ar [r]^-{ \widetilde{\mathscr {H}}_{\operatorname{\mathcal{C}}} } \ar [d] & \operatorname{\mathcal{S}}_{\ast } \ar [d]^{U} \\ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \ar [r]^-{ \mathscr {H}_{\operatorname{\mathcal{C}}} } & \operatorname{\mathcal{S}}. } \]

To define the upper horizontal map, we will use a variant of Construction

Construction Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote its homotopy coherent nerve. Let $J$ be a linearly ordered set and let $\overline{J}$ denote its opposite; for each element $j \in J$, we write $\overline{j}$ for the corresponding element of $\overline{J}$. Suppose we are given a morphism of simplicial sets $\sigma : \operatorname{N}_{\bullet }(J) \rightarrow \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) )$, which we identify with a simplicial functor $f: \operatorname{Path}[ \overline{J} \star J ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ (see Warning and Proposition Note that the composition

\[ \operatorname{N}_{\bullet }(J) \xrightarrow {\sigma } \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \xrightarrow { \mathscr {H}_{\operatorname{\mathcal{C}}} } \operatorname{\mathcal{S}} \]

can be identified with a simplicial functor $F_{\sigma }: \operatorname{Path}[J]_{\bullet } \rightarrow \operatorname{Kan}$, given on objects by the formula $F_{\sigma }(j) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( f( \overline{j} ), f(j) )_{\bullet }$ (see Proposition Let $J^{\triangleleft } = \{ x\} \star J$ denote the linearly ordered set obtained from $J$ by adding a new smallest element $x$. We extend $F_{\sigma }$ to a simplicial functor $\widetilde{F}_{\sigma }: \operatorname{Path}[ J^{\triangleleft } ]_{\bullet } \rightarrow \operatorname{Kan}$ as follows:


The functor $\widetilde{F}_{\sigma }$ carries the element $x \in J^{\triangleleft }$ to the Kan complex $\Delta ^0$.


Let $j$ be an element of $J$. Let us identify $\operatorname{Hom}_{ \operatorname{Path}[ J^{\triangleleft } ] }( x, j)_{\bullet }$ with the nerve $\operatorname{N}_{\bullet }( Q)$, where $Q$ is the collection of finite subsets $I \subseteq J$ satisfying $\mathrm{max}(I) = j$ (partially ordered by reverse inclusion). Similarly, we identify $\operatorname{Hom}_{ \operatorname{Path}[ \overline{J} \star J] }( \overline{j}, j )_{\bullet }$ with the nerve $\operatorname{N}_{\bullet }( Q' )$, where $Q'$ is the collection of finite subsets $I' \subseteq \overline{J} \star J$ satisfying $\mathrm{max}(I') = j$ and $\mathrm{min}(I') = \overline{j}$ (partially ordered by reverse inclusion). Then $\widetilde{F}_{\sigma }$ is defined on the morphism space $\operatorname{Hom}_{ \operatorname{Path}[ J^{\triangleleft } ] }(x,j)_{\bullet }$ by the composition

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Path}[ J^{\triangleleft } ] }(x,j)_{\bullet } & \simeq & \operatorname{N}_{\bullet }(Q) \\ & \xrightarrow { I \mapsto \overline{I} \cup I } & \operatorname{N}_{\bullet }(Q') \\ & \simeq & \operatorname{Hom}_{ \operatorname{Path}[ \overline{J} \star J] }( \overline{j}, j )_{\bullet } \\ & \xrightarrow {f} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( f(\overline{j}), f(j) )_{\bullet } \\ & \simeq & \operatorname{Fun}( \widetilde{F}_{\sigma }(x), \widetilde{F}_{\sigma }(j) ). \end{eqnarray*}

In the special case where $J$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $, we can identify $\widetilde{F}_{\sigma }$ with an $n$-simplex of the $\infty $-category of pointed spaces $\operatorname{\mathcal{S}}_{\ast } = \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})_{ \Delta ^0 / }$. The assignment $\sigma \mapsto \widetilde{F}_{\sigma }$ depends functorially on $[n]$, and therefore determines a functor $\widetilde{\mathscr {H}}_{\operatorname{\mathcal{C}}}: \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$. By construction, this functor fits into a commutative diagram

\begin{equation} \begin{gathered}\label{equation:witness-to-strict-Hom} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) ) \ar [r]^-{ \widetilde{\mathscr {H}}_{\operatorname{\mathcal{C}}} } \ar [d] & \operatorname{\mathcal{S}}_{\ast } \ar [d]^{U} \\ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \ar [r]^-{ \mathscr {H}_{\operatorname{\mathcal{C}}} } & \operatorname{\mathcal{S}}, } \end{gathered} \end{equation}

where the left vertical map is the twisted arrow fibration of Proposition and the right vertical map is the forgetful functor.

Exercise Verify that Construction is well-defined. That is, for every linearly ordered set $J$ and every morphism $\sigma : \operatorname{N}_{\bullet }(J) \rightarrow \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) )$, show that the simplicial functor $F_{\sigma }$ admits a unique extension $\widetilde{F}_{\sigma }: \operatorname{Path}[J^{\triangleleft }]_{\bullet } \rightarrow \operatorname{Kan}$ which satisfies conditions $(a)$ and $(b)$.

Proposition is an immediate consequence of the following more precise result:

Proposition Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. Then the diagram (8.18) is a categorical pullback square.

Proof. Note that the vertical maps in the diagram (8.18) are left fibrations (Propositions and It will therefore suffice to show that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of fibers

\begin{eqnarray*} \widetilde{\mathscr {H}}_{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{E}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{E}}) \times _{\operatorname{\mathcal{E}}} \{ Y\} & \rightarrow & \{ \mathscr {H}_{\operatorname{\mathcal{C}}}(X,Y) \} \times _{ \operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}_{\ast } \\ & = & \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{S}}}( \Delta ^0, \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet } ) \end{eqnarray*}

is a homotopy equivalence of Kan complexes (see Corollary Note that the coslice inclusion of Construction induces a monomorphism of simplicial sets $\iota : \operatorname{Hom}_{\operatorname{\mathcal{E}}}^{\mathrm{L}}(X,Y) \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{E}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{E}}) \times _{\operatorname{\mathcal{E}}} \{ Y\} $. Unwinding the definitions, we see that the composite map

\[ ( \widetilde{\mathscr {H}}_{X,Y} \circ \iota ): \operatorname{Hom}_{\operatorname{\mathcal{E}}}^{\mathrm{L}}(X,Y) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{S}}}( \Delta ^0, \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet } ) \]

coincides with isomorphism described in Remark It will therefore suffice to show that $\iota $ is a homotopy equivalence, which is a special case of Corollary $\square$