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

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 (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 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, 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 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

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 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 This assertion also has a counterpart in the setting of $\infty $-categories (Proposition, 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 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 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

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 Using the results of §8.2, we give an alternative characterization of representable profunctors (Proposition and show that they satisfy a universal mapping property (Corollary 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

Warning 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 In §8.3.6, we show that $\mathscr {H}$ is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{\mathcal{C}}$ (Proposition Our proof uses a recognition principle for $\operatorname{Hom}$-functors, which we formulate and prove in §8.3.5.


  • 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