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1.4.3 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 Let $S_{\bullet }$ and $T_{\bullet }$ be simplicial sets. Then the construction

\[ ([n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times S_{\bullet }, T_{\bullet } ) \]

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 $\operatorname{Fun}( S_{\bullet }, T_{\bullet } )$.

Note that, given an $n$-simplex $f$ of $\operatorname{Fun}( S_{\bullet }, T_{\bullet } )$ and an $n$-simplex $\sigma $ of $S_{\bullet }$, we can construct an $n$-simplex $\operatorname{ev}( f, \sigma )$ of $T_{\bullet }$, given by the composition

\[ \Delta ^ n \xrightarrow {\delta } \Delta ^ n \times \Delta ^ n \xrightarrow { \operatorname{id}\times \sigma } \Delta ^ n \times S_{\bullet } \xrightarrow { f } T_{\bullet }. \]

This construction determines a map of simplicial sets $\operatorname{ev}: \operatorname{Fun}( S_{\bullet }, T_{\bullet } ) \times { S_{\bullet } } \rightarrow T_{\bullet }$, which we will refer to as the evaluation map.

Proposition Let $S_{\bullet }$, $T_{\bullet }$, and $U_{\bullet }$ be simplicial sets. Then the composite map

\begin{eqnarray*} \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( U_{\bullet }, \operatorname{Fun}( S_{\bullet }, T_{\bullet } ) ) & \rightarrow & \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( U_{\bullet } \times S_{\bullet }, \operatorname{Fun}( S_{\bullet }, T_{\bullet } ) \times S_{\bullet } )\\ & \xrightarrow { \operatorname{ev}\circ } & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( U_{\bullet } \times S_{\bullet }, T_{\bullet } ) \end{eqnarray*}

is bijective.

Proof. Let $f: U_{\bullet } \times S_{\bullet } \rightarrow T_{\bullet }$ be a map of simplicial sets. For each $n$-simplex $\sigma $ of $U_{\bullet }$, the composite map

\[ \Delta ^ n \times S_{\bullet } \xrightarrow { \sigma \times \operatorname{id}} U_{\bullet } \times S_{\bullet } \xrightarrow {f} T_{\bullet } \]

can be regarded as an $n$-simplex of $\operatorname{Fun}( S_{\bullet }, T_{\bullet })$, which we will denote by $g(\sigma )$. The construction $\sigma \mapsto g(\sigma )$ determines a map of simplicial sets $g: U_{\bullet } \rightarrow \operatorname{Fun}( S_{\bullet }, T_{\bullet } )$. 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 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 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, 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 $\square$

Passing to homotopy categories, we obtain the following weaker result:

Corollary 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 as follows:

Corollary Let $S_{\bullet }$ be a simplicial set having homotopy category $\mathrm{h} \mathit{S}_{\bullet }$. Then, for any category $\operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{S}_{\bullet }, \operatorname{\mathcal{D}}) ) \times S_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Fun}(\mathrm{h} \mathit{S}_{\bullet }, \operatorname{\mathcal{D}}) ) \times \operatorname{N}_{\bullet }(\mathrm{h} \mathit{S}_{\bullet }) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}(\mathrm{h} \mathit{S}_{\bullet }, \operatorname{\mathcal{D}}) \times \mathrm{h} \mathit{S}_{\bullet }) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]

induces an isomorphism of simplicial sets $\rho _{S_{\bullet }}: \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{S}_{\bullet }, \operatorname{\mathcal{D}}) ) \simeq \operatorname{Fun}( S_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )$.

Proof. The construction $S_{\bullet } \mapsto \rho _{S_{\bullet }}$ 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 the category $\operatorname{Set_{\Delta }}$ is generated under colimits by objects of the form $\Delta ^ n$ (Lemma, it will suffice to prove Corollary in the special case where $S_{\bullet } \simeq \Delta ^ n$. In this case, the desired result follows from Proposition, since $S_{\bullet }$ is isomorphic to the nerve of the category $\operatorname{\mathcal{C}}= [n]$. $\square$

Corollary The formation of homotopy categories determines a functor $\operatorname{Set_{\Delta }}\rightarrow \operatorname{Cat}$ which commutes with finite products.

Proof. Since the construction $S_{\bullet } \mapsto \mathrm{h} \mathit{S}_{\bullet }$ preserves final objects, it will suffice to show that for any pair of simplicial sets $S_{\bullet }$ and $T_{\bullet }$, the canonical map

\[ u: \mathrm{h} \mathit{( S_{\bullet } \times T_{\bullet } )} \rightarrow \mathrm{h} \mathit{S}_{\bullet } \times \mathrm{h} \mathit{T}_{\bullet } \]

is an isomorphism of categories. In other words, we wish to show that for any category $\operatorname{\mathcal{C}}$, composition with $u$ induces a bijection

\[ \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S}_{\bullet } \times \mathrm{h} \mathit{T}_{\bullet }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{(S_{\bullet } \times T_{\bullet })}, \operatorname{\mathcal{C}}). \]

Unwinding the definitions, we see that this map is given by the composition

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S}_{\bullet } \times \mathrm{h} \mathit{T}_{\bullet }, \operatorname{\mathcal{C}}) & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S}_{\bullet }, \operatorname{Fun}( \mathrm{h} \mathit{T}_{\bullet }, \operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{T}_{\bullet }, \operatorname{\mathcal{C}}) ) ) \\ & \xrightarrow {\rho _{T_{\bullet }} \circ } & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{Fun}( T_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet } \times T_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{ (S_{\bullet } \times T_{\bullet } )}, \operatorname{\mathcal{C}}), \end{eqnarray*}

where $\rho _{T_{\bullet }}$ is the isomorphism appearing in the statement of Corollary $\square$

We will be primarily interested in the special case of Construction where the target simplicial set $T_{\bullet }$ is an $\infty $-category. In this case, we have the following result:

Theorem Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then the simplicial set $\operatorname{Fun}( S_{\bullet }, \operatorname{\mathcal{D}})$ is an $\infty $-category.

The proof of Theorem will require some combinatorial preliminaries; we defer the proof to ยง1.4.6.

Definition Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. It follows from Theorem 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 $\operatorname{\mathcal{D}}$.

Remark Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. By definition, the objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ can be identified with functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$, in the sense of Definition (that is, with maps of simplicial sets from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$).

Remark Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and suppose we are given a pair of functors $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. We define a natural transformation from $F$ to $G$ to be a map of simplicial sets $u: \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $u|_{ \{ 0\} \times \operatorname{\mathcal{C}}} = F$ and $u|_{ \{ 1\} \times \operatorname{\mathcal{C}}} = G$. In other words, a natural transformation from $F$ to $G$ is a morphism from $F$ to $G$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

Remark Let us abuse notation by identifying each ordinary category $\operatorname{\mathcal{E}}$ with the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$. In this case, Corollary implies that when $\operatorname{\mathcal{C}}$ is an $\infty $-category and $\operatorname{\mathcal{D}}$ is an ordinary category, then we have a canonical isomorphism $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$. In particular, the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is an ordinary category.