8.3.3 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.
Definition 8.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A $\operatorname{Hom}$-funtor for $\operatorname{\mathcal{C}}$ is a profunctor
\[ \mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}} \]
which is a covariant transport representation for twisted arrow coupling $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.
Proposition 8.3.3.2 (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 8.3.5.3 with Corollary 5.7.0.6 (applied to the left fibration $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$).
$\square$
Example 8.3.3.4. Let $\operatorname{\mathcal{C}}$ be a category. 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 $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. For a more general statement, see Proposition 8.3.6.2.
Variant 8.3.3.5. 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.
Notation 8.3.3.7. 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: Remark 8.3.3.3 guarantees the existence of homotopy equivalences $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \simeq \mathscr {H}(X,Y)$, which can be chosen to depend functorially on $X$ and $Y$ (as morphisms in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). Consequently, we can always modify the choice of $\operatorname{Hom}$-functor $\mathscr {H}$ to arrange that definitions $(a)$ and $(b)$ coincide (see Corollary 4.4.5.3).
Proposition 8.3.3.8. 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 $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. Then $\mathscr {H}$ is a balanced profunctor (see Definition 8.3.2.18).
Proof.
By virtue of Remark 8.3.2.19, it suffices to observe that the twisted arrow coupling $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ is balanced; see Example 8.2.6.2.
$\square$
Definition 8.3.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let
\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto h_ Y \]
be a functor. We say that $h_{\bullet }$ is a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ if the construction $(X,Y) \mapsto h_{Y}(X)$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, in the sense of Definition 8.3.3.1. Similarly, we say that a functor
\[ h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto h^{X} \]
is a contravariant Yoneda embedding for $\operatorname{\mathcal{C}}$ if the construction $(X,Y) \mapsto h^{X}(Y)$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.
If these conditions are satisfied, then the functors $h_{\bullet }$ and $h^{\bullet }$ are uniquely determined up to isomorphism. Moreover, for every object $X \in \operatorname{\mathcal{C}}$, the functor $h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable by $X$, and the functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$ (Proposition 8.3.5.5).
Variant 8.3.3.12. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the $\infty $-category of $\kappa $-small spaces (see Variant 5.6.4.12). For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $\kappa $-small.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits a covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits a contravariant Yoneda embedding $h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
See Variant 8.3.5.7.
Our main goal in this section is to prove the following:
Theorem 8.3.3.13 (Yoneda's Lemma for $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. Then the covariant and contravariant Yoneda embeddings
\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \]
are fully faithful functors, whose essential images are the full subcategories
\[ \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad \operatorname{Fun}^{\mathrm{corep}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \]
spanned by the representable and corepresentable functors, respectively.
Proof.
By virtue of Corollary 8.3.2.20, this is a reformulation of Proposition 8.3.3.8.
$\square$
We close this section by recording a simple observation about the Yoneda embedding.
Proposition 8.3.3.14. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Suppose we are given a diagram $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, where $K$ is a small simplicial set. The following conditions are equivalent:
- $(1)$
The morphism $\overline{f}$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(2)$
The composition $h_{\bullet } \circ \overline{f}$ is a limit diagram in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.
Following the convention of Remark 5.4.0.5, we can regard Proposition 8.3.3.14 as a special case of the following more precise assertion (applied in the special case where $\kappa = \lambda $ is a strongly inaccessible cardinal):
Variant 8.3.3.15. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be a locally $\lambda $-small $\infty $-category, and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda })$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Let $\kappa = \mathrm{ecf}(\lambda )$ be the exponential cofinality of $\lambda $, let $K$ be a $\kappa $-small simplicial set, and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the following conditions are equivalent:
- $(1)$
The morphism $\overline{f}$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(2)$
The composition $h_{\bullet } \circ \overline{f}$ is a limit diagram in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda })$.
Proof.
Since $K$ is $\kappa $-small, the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ admits $K$-indexed limits (Example 7.6.7.4). For each object $X \in \operatorname{\mathcal{C}}$, let $\operatorname{ev}_{X}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ denote the functor given by evaluation at $X$. By virtue of Proposition 7.1.6.1, condition $(2)$ is equivalent to the requirement that for each object $X \in \operatorname{\mathcal{C}}$, the composition
\[ K^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}},\operatorname{\mathcal{S}}^{< \lambda }) \xrightarrow { \operatorname{ev}_{X} } \operatorname{\mathcal{S}}^{< \lambda } \]
is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$. Since the composite functor $(\operatorname{ev}_{X} \circ h_{\bullet }): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ is corepresentable by $X$, the equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 7.4.5.13 (and Remark 7.4.5.15).
$\square$
Corollary 8.3.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be an infinite cardinal. Then there exists a fully faithful functor $F: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$, where $\widehat{\operatorname{\mathcal{C}}}$ admits $\kappa $-small limits and colimits. Moreover, we can arrange that $F$ preserves the limits of all $\kappa $-small diagrams which exist in $\operatorname{\mathcal{C}}$.
Proof.
Using Remark 5.4.3.19, we can choose an uncountable cardinal $\lambda $ of exponential cofinality $\geq \kappa $. Enlarging $\lambda $ if necessary, we may assume that $\operatorname{\mathcal{C}}$ is locally $\lambda $-small. Let $\widehat{\operatorname{\mathcal{C}}}$ denote the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$ and let $F = h_{\bullet }$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{S}}^{< \lambda }$ admits $\kappa $-small limits and colimits (Remark 7.4.5.7 and Variant 7.4.5.8), the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ has the same property (Corollary 7.1.6.2). Moreover, the functor $F$ is fully faithful (Theorem 8.3.3.13) and preserves limits of $\kappa $-small diagrams (Remark 8.3.3.16).
$\square$