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8.2.5 The Yoneda Embedding

We now use the results of 8.2.3 to construct an $\infty $-categorical analogue of the Yoneda embedding.

Definition 8.2.5.1. 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 8.2.3.2. 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 $\operatorname{\mathcal{C}}$.

Remark 8.2.5.2 (Duality). A functor $h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ is a contravariant Yoneda embedding for $\operatorname{\mathcal{C}}$ if and only if it is a covariant Yoneda embedding for the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$; see Remark 8.2.3.7.

Remark 8.2.5.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. By virtue of Proposition 8.2.3.10, the following conditions are equivalent:

  • The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small.

  • The $\infty $-category $\operatorname{\mathcal{C}}$ admits a covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.

  • The $\infty $-category $\operatorname{\mathcal{C}}$ admits a contravariant Yoneda embedding $h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$.

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 8.2.5.6).

Variant 8.2.5.4. 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 8.2.3.13.

Our main goal in this section is to prove the following:

Theorem 8.2.5.5 (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.

By virtue of Corollary 8.2.2.12, Theorem 8.2.5.5 is equivalent to the assertion that the $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a balanced profunctor, in the sense of Definition 8.2.2.11. This is a consequence of the following more precise result:

Proposition 8.2.5.6. 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 object $X \in \operatorname{\mathcal{C}}$, we can evaluate $\alpha $ on the object $\operatorname{id}_{X} \in \operatorname{Tw}(\operatorname{\mathcal{C}})$ to obtain a vertex $\alpha (\operatorname{id}_ X) \in \mathscr {H}(X,X)$. The following conditions are equivalent:

$(1)$

The natural transformation $\alpha $ exhibit $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, in the sense of Remark 8.2.3.3.

$(2)$

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 $X \in \operatorname{\mathcal{C}}$ (in the sense of Definition 5.7.6.1).

$(3)$

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 $X \in \operatorname{\mathcal{C}}$ (in the sense of Variant 5.7.6.2).

Proof. We will show that $(1) \Leftrightarrow (2)$; the proof of the equivalence $(1) \Leftrightarrow (3)$ 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

8.19
\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 5.1.6.15, 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 5.7.6.21, 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). $\square$

Corollary 8.2.5.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(\mathscr {H}, \alpha )$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then the following conditions are equivalent:

$(1)$

The morphism $f$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$

$(2)$

The vertex $\alpha (f) \in \mathscr {H}(X,Y)$ exhibits the functor $\mathscr {H}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by $Y$.

$(3)$

The vertex $\alpha (f) \in \mathscr {H}(X,Y)$ exhibits the functor $\mathscr {H}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by $X$.

Corollary 8.2.5.8. 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 from $\operatorname{\mathcal{C}}$ to itself.

Proof. By virtue of Proposition 8.2.5.6, the profunctor $\mathscr {H}$ is both representable and corepresentable. Fix a pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and a vertex $\eta \in \mathscr {H}(X,Y)$. We wish to show that $\eta $ exhibits the functor $\mathscr {H}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by $X$ if and only if it exhibits the functor $\mathscr {H}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by $Y$. Both conditions depend only on the image of $\eta $ in $\pi _0( \mathscr {H}(X,Y) )$. We may therefore assume without loss of generality that $\eta = \alpha (f)$, where $\alpha : \underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ is a natural transformation which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, and $f: X \rightarrow Y$ is a morphism in the $\infty $-category $\operatorname{\mathcal{C}}$. In this case, the equivalence follows from Corollary 8.2.5.7. $\square$

We close this section by recording a simple observation about the Yoneda embedding.

Proposition 8.2.5.9. 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 8.2.5.9 as a special case of the following more precise assertion (applied in the special case where $\kappa = \lambda $ is a strongly inaccessible cardinal):

Variant 8.2.5.10. 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). $\square$

Remark 8.2.5.11. In the situation of Variant 8.2.5.10, suppose that the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits. Then the $\infty $-category of representable functors $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{ < \lambda } )$ also admits $K$-indexed limits, which are preserved by the inclusion functor $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{ < \lambda } ) \hookrightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{ < \lambda } )$.