Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

8.2.3 Hom-Functors for $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In ยง4.6.1, we associated to every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ a Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ parametrizing morphisms from $X$ to $Y$. In this section, we will promote the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to a functor of $\infty $-categories.

Notation 8.2.3.1. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.6.1.1). We will regard the contractible Kan complex $\Delta ^0$ as an object of $\operatorname{\mathcal{S}}$. For every $\infty $-category $\operatorname{\mathcal{E}}$, we let $\underline{ \Delta ^0}_{\operatorname{\mathcal{E}}}$ denote the constant functor $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$ taking the value $\Delta ^0$.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{Tw}(\operatorname{\mathcal{C}})$ denote its twisted arrow $\infty $-category (Construction 8.1.1.1). For any profunctor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, we let $\mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{S}}$ denote the composition of $\mathscr {H}$ with the left fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of Proposition 8.1.1.10.

Definition 8.2.3.2 ($\operatorname{Hom}$-Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\lambda : \operatorname{Tw}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ be the left fibration of Proposition 8.1.1.10. A $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ is a pair $( \mathscr {H}, \alpha )$, where $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a functor and $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ is a natural transformation which 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

\[ \alpha _{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {H}(X,Y) ). \]

Remark 8.2.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often abuse terminology by referring to a functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if there exists a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{\operatorname{Tw}(\operatorname{\mathcal{C}})}$ which satisfies condition $(\ast )$ of Definition 8.2.3.2. In this case, we will say that $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.

Remark 8.2.3.4. 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 functor, and let $\alpha : \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ be a natural transformation. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Notation 8.1.2.10 and Remark 5.6.1.5 supply canonical isomorphisms

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \quad \quad \mathscr {H}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {H}(X,Y) ) \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Consequently, the homotopy class of the morphism $\alpha _{X,Y}$ appearing in Definition 8.2.3.2 can be identified with a map $[ \alpha _{X,Y} ]: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \mathscr {H}(X,Y)$ in $\mathrm{h} \mathit{\operatorname{Kan}}$. The natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if and only if $[ \alpha _{X,Y} ]$ is an isomorphism for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$.

Example 8.2.3.5. Let $\operatorname{\mathcal{C}}$ be a (locally small) category. Then the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ determines a functor

\[ \mathscr {H}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set}) \subset \operatorname{\mathcal{S}}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), \]

which is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. More precisely, there is a natural transformation $\alpha : \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}))}$ which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, given explicitly by assigning to each object $(f: X \rightarrow Y)$ of $\operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$ the inclusion map $\{ f\} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \mathscr {H}(X,Y)$.

Remark 8.2.3.6 (Naturality). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denotes its homotopy category, which we view as enriched over the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Construction 4.6.8.13). The enrichment determines a functor

\[ \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(-, -): \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}, \]

given on objects by the formula $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. Suppose we are given a functor of $\infty $-categories $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$. Passing to homotopy categories, we obtain a functor

\[ \mathrm{h} \mathit{\mathscr {H}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}. \]

For any natural transformation $\alpha : \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$, the construction $X,Y \mapsto [ \alpha _{X,Y} ]$ described in Remark 8.2.3.4 determines a natural transformation from $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(-, -)$ to the functor $\mathrm{h} \mathit{\mathscr {H}}$ (see Corollary 8.1.2.13). This natural transformation is an isomorphism if and only if $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.

Remark 8.2.3.7 (Duality). 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 functor, and let $\mathscr {H}': \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be the functor obtained from $\mathscr {H}$ by transposing its arguments. If $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, then $\mathscr {H}'$ is a $\operatorname{Hom}$-functor for the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. More precisely, if $\alpha : \underline{ \Delta ^{0} }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, then it also exhibits $\mathscr {H}'$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (by means of the identification $\operatorname{Tw}(\operatorname{\mathcal{C}}) \simeq \operatorname{Tw}( \operatorname{\mathcal{C}}^{\operatorname{op}} )$ supplied by Remark 8.1.1.6).

Remark 8.2.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor. The datum of a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{\operatorname{Tw}(\operatorname{\mathcal{C}})}$ can be identified with a commutative diagram of $\infty $-categories

8.17
\begin{equation} \begin{gathered}\label{diagram:Hom-witness} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] \ar [d] & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}\ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {H} } & \operatorname{\mathcal{S}}. } \end{gathered} \end{equation}

In this case, the natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if and only if the diagram (8.17) is a categorical pullback square (see Corollary 5.1.6.15).

Remark 8.2.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Using Remark 8.2.3.8, we see that a functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if and only if it is a covariant transport representation for the left fibration $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$. In other words, $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if $\lambda $ can be lifted to an equivalence of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ with the $\infty $-category of elements $\int _{ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}} \mathscr {H}$.

Proposition 8.2.3.10 (Existence and Uniqueness). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ if and only if it is locally small. If this condition is satisfied, then $\mathscr {H}$ is uniquely determined up to isomorphism.

Proof. Combine Remark 8.2.3.9 with Corollary 5.7.0.6 (applied to the left fibration $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$). $\square$

Variant 8.2.3.11. Let $\kappa $ be an uncountable cardinal, and let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the $\infty $-category of $\kappa $-small spaces (Variant 5.6.4.12). Then an $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\operatorname{Hom}$-functor

\[ \mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \]

if and only if it is locally $\kappa $-small. If this condition is satisfied, then $\mathscr {H}$ is uniquely determined up to isomorphism.

For many applications, the uniqueness assertion of Proposition 8.2.3.10 is insufficiently precise. When viewed abstractly as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$, a $\operatorname{Hom}$-functor $\mathscr {H}$ is uniquely determined up to isomorphism but not up to canonical isomorphism. We can remedy the situation by considering the additional datum of a natural transformation $\alpha : \underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. In this case, the pair $( \mathscr {H}, \alpha )$ is unique up to a contractible choice, when viewed as an object of the $\infty $-category $\{ \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \} \operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$. This is a consequence of the following:

Proposition 8.2.3.12. 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 8.2.3.2.

$(2)$

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 7.6.2.15 and Remark 8.2.3.8. Since $\operatorname{\mathcal{C}}$ is locally small, Proposition 8.2.3.10 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 8.2.3.13. 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 8.2.3.12, we can replace $\operatorname{\mathcal{S}}$ with the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ of $\kappa $-small spaces (Variant 5.6.4.12).

Notation 8.2.3.14. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. We will often use the notation $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$ to denote a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$. Beware that this convention introduces a slight potential for confusion. Given a pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have two potentially different definitions of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$:

$(a)$

The Kan complex $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $ of Construction 4.6.1.1, which is well-defined up to canonical isomorphism.

$(b)$

The Kan complex $\mathscr {H}(X,Y)$, which is only well-defined up to homotopy equivalence (since it depends on a choice of $\operatorname{Hom}$-functor $\mathscr {H}$).

However, the danger is slight: if we choose a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, then Remark 8.2.3.4 supplies canonical isomorphisms $[ \alpha _{X,Y} ]: \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \mathscr {H}(X,Y)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Consequently, if each of the Kan complexes $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $ is small, we can modify the choice of $\operatorname{Hom}$-functor $\mathscr {H}$ to arrange that definitions $(a)$ and $(b)$ coincide (see Corollary 4.4.5.3).