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8.3 The Yoneda Embedding

Let $\operatorname{\mathcal{C}}$ be a category. For every object $X \in \operatorname{\mathcal{C}}$, we let $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor corepresented by $X$, given on objects by the formula $h^{X}(Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. The construction $X \mapsto h^{X}$ determines a functor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to the functor category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set})$, which we refer to as the (contravariant) Yoneda embedding. This terminology is justified by the following:

Proposition 8.3.0.1 (Yoneda's Lemma, Weak Form). For any (locally small) category $\operatorname{\mathcal{C}}$, the Yoneda embedding

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \quad \quad X \mapsto h^ X \]

is fully faithful.

The goal of this section is to extend Proposition 8.3.0.1 to the setting of $\infty $-categories. Our first step is to construct an analogue of the functor $X \mapsto h^{X}$. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a Kan complex (Proposition 4.6.1.10), which we can regard as an object of the $\infty $-category $\operatorname{\mathcal{S}}$. In §8.3.3, we show that the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be upgraded to a functor from the product $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ to the $\infty $-category $\operatorname{\mathcal{S}}$. More precisely, every locally small $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, which is characterized (up to isomorphism) by the requirement that it is a covariant transport representation for the twisted arrow fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of Proposition 8.3.3.2. This condition guarantees that for every object $X \in \operatorname{\mathcal{C}}$, the functor $\mathscr {H}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$. We can therefore identify $\mathscr {H}$ with a functor

\[ h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {H}(X, -) \]

carrying each object of $\operatorname{\mathcal{C}}$ to a functor that it corepresents; we will refer to $h^{\bullet }$ as a contravariant Yoneda embedding for $\operatorname{\mathcal{C}}$ (Definition 8.3.3.9).

To show that the Yoneda embedding is fully faithful, we will need an additional ingredient. Let us return to the situation where $\operatorname{\mathcal{C}}$ is an ordinary category. Proposition 8.3.0.1 asserts that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the natural map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) }( h^{X}, h^{Y} )$ is a bijection. It is easy to see that this map is injective: in fact, it has a left inverse $T: \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) }( h^{X}, h^{Y} ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, X )$, which carries a natural transformation $\alpha : h^{X} \rightarrow h^{Y}$ to the element $\alpha _{X}( \operatorname{id}_{X} ) \in h^{Y}(X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$. It will therefore suffice to show that $T$ is bijective. This is a consequence of the following strong version of Yoneda's lemma: for every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$, the evaluation map

\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) }( h^{X}, \mathscr {F} ) \rightarrow \mathscr {F}(X) \quad \quad \alpha \mapsto \alpha _{X}( \operatorname{id}_ X) \]

is a bijection (Proposition 8.3.1.1). This assertion also has a counterpart in the setting of $\infty $-categories (Proposition 8.3.1.3), which we formulate and prove in §8.3.1.

To exploit the universal mapping property of (co)representable functors, it will be convenient to introduce some terminology. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. We define a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ to be a functor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ (Definition 8.3.2.1). We say that a profunctor $\mathscr {K}$ is corepresentable if, for every object $X \in \operatorname{\mathcal{C}}$, the functor $\mathscr {K}(X, -): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable. In this case, the construction $X \mapsto \mathscr {K}(X,-)$ determines a functor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to the full subcategory $\operatorname{Fun}^{\mathrm{corep}}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors. In §8.3.2, we establish a criterion for this functor to be an equivalence of $\infty $-categories (Corollary 8.3.2.20). Our $\infty $-categorical version of Yoneda's lemma then follows by specializing this criterion to the situation where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{D}}$ and $\mathscr {K}$ is a $\operatorname{Hom}$-functor (Theorem 8.3.3.13).

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is locally small, so that it admits a $\operatorname{Hom}$-functor $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$. For every functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, the composition

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { \operatorname{id}\times G} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\xrightarrow {\mathscr {H}} \operatorname{\mathcal{S}}, \]

can be regarded as a profunctor $\mathscr {K}_{G}$ from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$, given informally by the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( X, G(Y) )$. In §8.3.4, we show that this construction determines a fully faithful functor $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$, whose essential image is spanned by the representable profunctors (Proposition 8.3.4.1). Using the results of §8.2, we give an alternative characterization of representable profunctors (Proposition 8.3.4.14) and show that they satisfy a universal mapping property (Corollary 8.3.4.20). As an application, we show that morphism spaces in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ can be computed as limits indexed by the twisted arrow $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{D}})$ (Example 8.3.4.21).

Warning 8.3.0.2. If $\operatorname{\mathcal{C}}$ is an ordinary category, the Yoneda embedding

\[ h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \hookrightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \quad \quad X \mapsto h^{X} \]

is given by a completely explicit construction. Beware that in the $\infty $-categorical setting, the Yoneda embedding depends on a choice of covariant transport representation for the twisted arrow fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$, which is well-defined only up to isomorphism. However, it is sometimes possible to eliminate this ambiguity. Suppose that $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_0 )$ is the homotopy coherent nerve of a locally Kan simplicial category $\operatorname{\mathcal{C}}_0$. In this case, the simplicial enrichment of $\operatorname{\mathcal{C}}_0$ determines a functor of simplicial categories

\[ \operatorname{\mathcal{C}}_0^{\operatorname{op}} \times \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Kan}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}(X,Y)_{\bullet }. \]

Passing to the homotopy coherent nerve, we obtain a functor of $\infty $-categories $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ (Construction 8.3.6.1). In §8.3.6, we show that $\mathscr {H}$ is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{\mathcal{C}}$ (Proposition 8.3.6.2). Our proof uses a recognition principle for $\operatorname{Hom}$-functors, which we formulate and prove in §8.3.5.

Structure

  • Subsection 8.3.1: Yoneda's Lemma
  • Subsection 8.3.2: Profunctors of $\infty $-Categories
  • Subsection 8.3.3: Hom-Functors for $\infty $-Categories
  • Subsection 8.3.4: Representable Profunctors
  • Subsection 8.3.5: Recognition of Hom-Functors
  • Subsection 8.3.6: Strict Models for Hom-Functors