8.3.5 Recognition of Hom-Functors
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $\operatorname{\mathcal{C}}$ is locally small, then Proposition 8.3.3.2 guarantees that it admits a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, which is uniquely determined up to isomorphism. Our goal in this section is to formulate a more precise statement, which characterizes the functor $\mathscr {H}$ up to canonical isomorphism (see Proposition 8.3.5.6).
Definition 8.3.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ denote the left fibration of Proposition 8.1.1.11, and let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. We say that a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if it satisfies the following condition:
- $(\ast )$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the natural transformation $\alpha $ induces a homotopy equivalence of Kan complexes
\[ \alpha _{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {H}(X,Y) ). \]
Proposition 8.3.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor, and let $\alpha : \underline{ \Delta ^{0} }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ be a natural transformation. The following conditions are equivalent:
- $(1)$
The natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.3.5.1).
- $(2)$
The natural transformation $\alpha $ exhibits the profunctor $\mathscr {H}$ as represented by the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 8.3.4.12). That is, for every object $X \in \operatorname{\mathcal{C}}$, the vertex $\alpha ( \operatorname{id}_{X} ) \in \mathscr {H}(X,X)$ exhibits the functor $\mathscr {H}(-, X): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by the object $X$.
- $(3)$
The natural transformation $\alpha $ exhibits the profunctor $\mathscr {H}$ as corepresented by the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Variant 8.3.4.16). That is, for every object $X \in \operatorname{\mathcal{C}}$, the vertex $\alpha ( \operatorname{id}_{X} ) \in \mathscr {H}(X,X)$ exhibits the functor $\mathscr {H}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by the object $X$.
Proof.
We will show that $(1) \Leftrightarrow (3)$; the proof of the equivalence $(1) \Leftrightarrow (2)$ is similar. The natural transformation $\alpha $ can be identified with a functor $T: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}$. For each object $X \in \operatorname{\mathcal{C}}$, let $T_{X}$ denote the restriction of $B$ to the simplicial subset $\{ X \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Tw}(\operatorname{\mathcal{C}})$, and consider the following condition:
- $(1_ X)$
The diagram of $\infty $-categories
8.57
\begin{equation} \begin{gathered}\label{diagram:Hom-witness-later} \xymatrix@R =50pt@C=50pt{ \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T_ X} \ar [d] & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}\ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {H}(X, -) } & \operatorname{\mathcal{S}}} \end{gathered} \end{equation}
is a categorical pullback square.
By virtue of Corollary 5.1.7.15, the natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if and only if it satisfies condition $(1_ X)$ for every object $X \in \operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that $(1_ X)$ is satisfied if and only if $\alpha (\operatorname{id}_ X) \in \mathscr {H}(X,X)$ exhibits the functor $\mathscr {H}(X, -)$ as corepresented by $X$. This is a special case of Proposition 5.6.6.21, since the $\operatorname{id}_{X}$ is an initial object of the $\infty $-category $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ (Proposition 8.1.2.1).
$\square$
Proposition 8.3.5.6. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category, let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor, and let $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ be a natural transformation. The following conditions are equivalent:
- $(1)$
The natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$: that is, it satisfies condition $(\ast )$ of Definition 8.3.5.1.
- $(2)$
The diagram
\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [dr]_-{\mathscr {H}} \ar@ {<=}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [ur]^{\lambda } \ar [rr]_{ \underline{ \Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} } & & \operatorname{\mathcal{S}}. } \]
exhibits $\mathscr {H}$ as a left Kan extension of the constant functor $\underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) }$ along the left fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.
- $(3)$
The pair $( \mathscr {H}, \alpha )$ is initial when viewed as an object of the oriented fiber product $\{ \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \} \operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 7.6.2.17 and Remark 8.3.5.2. Since $\operatorname{\mathcal{C}}$ is locally small, Proposition 8.3.3.2 guarantees that the functor $\underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) }$ admits a left Kan extension along $\lambda $, so the equivalence $(2) \Leftrightarrow (3)$ follows from Corollary 7.3.6.5.
$\square$
Variant 8.3.5.7. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small. Then, in the statement of Proposition 8.3.5.6, we can replace $\operatorname{\mathcal{S}}$ with the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ of $\kappa $-small spaces (Variant 5.5.4.12).
Corollary 8.3.5.8 (Functoriality of $\operatorname{Hom}$-Functors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. Choose natural transformations
\[ \alpha : \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}_{\operatorname{\mathcal{C}}}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}})} \quad \quad \beta : \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{D}}) } \rightarrow \mathscr {H}_{\operatorname{\mathcal{D}}}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} \]
which exhibit $\mathscr {H}_{\operatorname{\mathcal{C}}}$ and $\mathscr {H}_{\operatorname{\mathcal{D}}}$ as $\operatorname{Hom}$-functors for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. Then there exists a natural transformation $\gamma : \mathscr {H}_{\operatorname{\mathcal{C}}}(-,-)\rightarrow \mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), F(-) )$ for which the diagram
8.58
\begin{equation} \begin{gathered}\label{equation:Hom-functoriality} \xymatrix@R =50pt@C=50pt{ & \underline{ \Delta ^{0} }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \ar [dl]_-{ [\alpha ] } \ar [dr]^-{ [\beta ]} \\ \mathscr {H}_{\operatorname{\mathcal{C}}}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \ar [rr]^{ [\gamma ] } & & \mathscr {H}_{\operatorname{\mathcal{D}}}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } } \end{gathered} \end{equation}
commutes (in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}) }$). Moreover, the natural transformation $\gamma $ is uniquely determined up to homotopy.
Proof.
This is a special case of Proposition 7.3.6.1, since $\alpha $ exhibits $\mathscr {H}_{\operatorname{\mathcal{C}}}$ as a left Kan extension of $\underline{ \Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ along the left fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ (Proposition 8.3.5.6).
$\square$