1.5.3 The $\infty $-Category of Functors
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. Then we can form a new category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, whose objects are functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and whose morphisms are natural transformations. In this section, we describe an analogous construction in the setting of $\infty $-categories.
Construction 1.5.3.1. Let $S$ and $T$ be simplicial sets. Then the construction
\[ ([n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times S, T ) \]
determines a functor from the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the category of sets. We regard this functor as a simplicial set which we will denote by $\operatorname{Fun}( S, T)$.
Note that, given an $n$-simplex $f$ of $\operatorname{Fun}( S, T )$ and an $n$-simplex $\sigma $ of $S$, we can construct an $n$-simplex $\operatorname{ev}( f, \sigma )$ of $T$, given by the composition
\[ \Delta ^ n \xrightarrow {\delta } \Delta ^ n \times \Delta ^ n \xrightarrow { \operatorname{id}\times \sigma } \Delta ^ n \times S \xrightarrow { f } T. \]
This construction determines a map of simplicial sets $\operatorname{ev}: \operatorname{Fun}( S, T) \times { S} \rightarrow T$, which we will refer to as the evaluation map.
Proposition 1.5.3.2. Let $S$, $T$, and $U$ be simplicial sets. Then the composite map
\begin{eqnarray*} \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( U, \operatorname{Fun}( S, T ) ) & \rightarrow & \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( U \times S, \operatorname{Fun}( S, T ) \times S )\\ & \xrightarrow { \operatorname{ev}\circ } & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( U \times S, T ) \end{eqnarray*}
is bijective.
Proof.
Let $f: U \times S \rightarrow T$ be a map of simplicial sets. For each $n$-simplex $\sigma $ of $U$, the composite map
\[ \Delta ^ n \times S \xrightarrow { \sigma \times \operatorname{id}} U \times S \xrightarrow {f} T \]
can be regarded as an $n$-simplex of $\operatorname{Fun}( S, T)$, which we will denote by $g(\sigma )$. The construction $\sigma \mapsto g(\sigma )$ determines a map of simplicial sets $g: U \rightarrow \operatorname{Fun}( S, T )$. We leave as an exercise for the reader to verify that $g$ is the unique map satisfying $\theta (g) = f$.
$\square$
Beware that the notation of Construction 1.5.3.1 is potentially confusing, because it conflicts with our use of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ to denote the category of functors from a category $\operatorname{\mathcal{C}}$ to a category $\operatorname{\mathcal{D}}$. However, these usages are compatible:
Proposition 1.5.3.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories and let $e: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ denote the evaluation functor, given on objects by the formula $e(F,C) = F(C)$. Then the composite map
\[ \operatorname{N}_{\bullet }( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{N}_{\bullet }(e) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]
corresponds, under the bijection of Proposition 1.5.3.2, to an isomorphism of simplicial sets $\rho : \operatorname{N}_{\bullet }( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )$.
Proof.
For each $n \geq 0$, the map $\rho $ is given on $n$-simplices by the composition
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{N}_{\bullet }( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) ) & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( [n], \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Cat}}( [n] \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \\ & \xrightarrow {v} & \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( [n] \times \operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( [n] ) \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )). \end{eqnarray*}
It will therefore suffice to show that $v$ is bijective, which is a special case of Proposition 1.3.3.1.
$\square$
Passing to homotopy categories, we obtain the following weaker result:
Corollary 1.5.3.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. Then there is a canonical isomorphism of categories
\[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{Fun}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ). \]
We can also generalize Proposition 1.5.3.3 as follows:
Corollary 1.5.3.5. Let $S$ be a simplicial set having homotopy category $\mathrm{h} \mathit{S}$. Then, for any category $\operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{S}, \operatorname{\mathcal{D}}) ) \times S \rightarrow \operatorname{N}_{\bullet }( \operatorname{Fun}(\mathrm{h} \mathit{S}, \operatorname{\mathcal{D}}) ) \times \operatorname{N}_{\bullet }(\mathrm{h} \mathit{S}) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}(\mathrm{h} \mathit{S}, \operatorname{\mathcal{D}}) \times \mathrm{h} \mathit{S}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]
induces an isomorphism of simplicial sets $\rho _{S}: \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{S} \operatorname{\mathcal{D}}) ) \simeq \operatorname{Fun}( S, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )$.
Proof.
The construction $S \mapsto \rho _{S}$ carries colimits (in the category $\operatorname{Set_{\Delta }}$ of simplicial sets) to limits (in the category $\operatorname{Fun}([1], \operatorname{Set_{\Delta }})$ of morphisms between simplicial sets). Since every simplicial set can be realized as a colimit of standard simplices (Remark 1.1.3.13), it will suffice to prove Corollary 1.5.3.5 in the special case where $S = \Delta ^ n$ for some $n \geq 0$. In this case, the desired result follows from Proposition 1.5.3.3, since $S$ is isomorphic to the nerve of the category $\operatorname{\mathcal{C}}= [n]$.
$\square$
Corollary 1.5.3.6. The formation of homotopy categories determines a functor $\operatorname{Set_{\Delta }}\rightarrow \operatorname{Cat}$ which commutes with finite products.
Proof.
Since the construction $S \mapsto \mathrm{h} \mathit{S}$ preserves final objects, it will suffice to show that for any pair of simplicial sets $S$ and $T$, the canonical map $u: \mathrm{h} \mathit{( S \times T)} \rightarrow \mathrm{h} \mathit{S} \times \mathrm{h} \mathit{T}$. s an isomorphism of categories. In other words, we wish to show that for any category $\operatorname{\mathcal{C}}$, composition with $u$ induces a bijection
\[ \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S} \times \mathrm{h} \mathit{T}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{(S \times T)}, \operatorname{\mathcal{C}}). \]
Unwinding the definitions, we see that this map is given by the composition
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S} \times \mathrm{h} \mathit{T}, \operatorname{\mathcal{C}}) & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S}, \operatorname{Fun}( \mathrm{h} \mathit{T}, \operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{T}, \operatorname{\mathcal{C}}) ) ) \\ & \xrightarrow {\rho _{T} \circ } & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{Fun}( T, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S \times T, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{ (S \times T )}, \operatorname{\mathcal{C}}), \end{eqnarray*}
where $\rho _{T}$ is the isomorphism appearing in the statement of Corollary 1.5.3.5.
$\square$
We will be primarily interested in the special case of Construction 1.5.3.1 where the target simplicial set $T$ is an $\infty $-category. In this case, we have the following result:
Theorem 1.5.3.7. Let $S$ be a simplicial set and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then the simplicial set $\operatorname{Fun}( S, \operatorname{\mathcal{D}})$ is an $\infty $-category.
The proof of Theorem 1.5.3.7 will require some combinatorial preliminaries; we defer the proof to ยง1.5.6.
Definition 1.5.3.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. It follows from Theorem 1.5.3.7 that the simplicial set $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is also an $\infty $-category. We will refer to $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ as the $\infty $-category of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.