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8.2 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 8.2.0.1 (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 generalize Proposition 8.2.0.1 the $\infty $-categorical setting. Our first step is to construct an appropriate analogue of the Yoneda embedding for an $\infty $-category $\operatorname{\mathcal{C}}$. 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 4.6.1.9), which we can regard as an object of the $\infty $-category $\operatorname{\mathcal{S}}$. In §8.2.3, 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 twisted arrow fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ or Proposition 8.2.3.10. 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 8.2.5.1).

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 8.2.0.1 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 8.2.1.1). This assertion also has a counterpart in the setting of $\infty $-categories (Proposition 8.2.1.3), which we prove in §8.2.1.

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 8.2.2.1). 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 §8.2.2, we give necessary and sufficient conditions for this functor to be fully faithful (Proposition 8.2.2.10) or an equivalence of $\infty $-categories (Corollary 8.2.2.12). In §8.2.5 we specialize these results to the situation where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{D}}$ and $\mathscr {K}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, and show that the contravariant Yoneda embedding $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ is fully faithful (Theorem 8.2.5.5).

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 $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$. 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 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 §8.2.6, we show that this construction determines a fully faithful functor $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$ (Proposition 8.2.6.1). Similarly, if $\operatorname{\mathcal{D}}$ is locally small, then there is a fully faithful functor $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$, which carries a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ to the profunctor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), Y)$. In §8.2.7, we show that functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are adjoint if and only if the corresponding profunctors are isomorphic (Proposition 8.2.7.1). Stated more informally, $F$ is left adjoint to $G$ if it is possible to choose homotopy equivalences $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) )$ which depend functorially on $X$ and $Y$. To prove this, we exploit the fact that (co)representable profunctors can be characterized by a universal mapping property (Proposition 8.2.6.15).

Warning 8.2.0.2. 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. In §8.2.4, we study the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_0 )$ is the homotopy coherent nerve of a locally Kan simplicial category $\operatorname{\mathcal{C}}_0$. In this case, we give a concrete example of a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, obtained as the homotopy coherent nerve of the functor

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

determined by the simplicial enrichment of $\operatorname{\mathcal{C}}_0$ (see Proposition 8.2.4.2).

Structure

  • Subsection 8.2.1: Yoneda's Lemma
  • Subsection 8.2.2: Profunctors of $\infty $-Categories
  • Subsection 8.2.3: Hom-Functors for $\infty $-Categories
  • Subsection 8.2.4: Strict Models for Hom-Functors
  • Subsection 8.2.5: The Yoneda Embedding
  • Subsection 8.2.6: Representable Profunctors
  • Subsection 8.2.7: Adjunctions as Profunctors