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5.7.6 Application: Corepresentable Functors

Let $\operatorname{\mathcal{C}}$ be a category. Every object $X \in \operatorname{\mathcal{C}}$ determines a functor

\[ h^ X: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}\quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), \]

which we refer to as the functor corepresented by $X$. We say that a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{Set}$ is corepresentable if it is isomorphic to $h^ X$ for some object $X \in \operatorname{\mathcal{C}}$. Our goal in this section is to develop an $\infty $-categorical counterpart of the notion of corepresentable functor (and the dual notion of representable functor), where we replace the ordinary category $\operatorname{Set}$ by the $\infty $-category $\operatorname{\mathcal{S}}$ of Construction 5.6.1.1.

We begin with an elementary observation. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor between ordinary categories. For each object $X \in \operatorname{\mathcal{C}}$, Yoneda's lemma supplies a bijection

\[ \mathscr {F}(X) \xrightarrow {\sim } \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) }( h^ X, \mathscr {F} ). \]

Concretely, this bijection carries each element $x \in \mathscr {F}(X)$ to a natural transformation $\alpha _ x: h^ X \rightarrow \mathscr {F}$, characterized by the requirement that it carries each $Y \in \operatorname{\mathcal{C}}$ to the composite map

5.60
\begin{equation} \label{equation:map-from-Yoneda} h^ X(Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\mathscr {F}} \operatorname{Hom}_{\operatorname{Set}}( \mathscr {F}(X), \mathscr {F}(Y) ) \xrightarrow {\operatorname{ev}_ x} \mathscr {F}(Y). \end{equation}

The functor $\mathscr {F}$ is corepresentable if it is possible to choose the object $X \in \operatorname{\mathcal{C}}$ and the element $x \in \mathscr {F}(X)$ so that the map (5.60) is bijective, for each $Y \in \operatorname{\mathcal{C}}$. This motivates the following:

Definition 5.7.6.1 (Corepresentable Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor, and let $x$ be a vertex of the Kan complex $\mathscr {F}(X)$. We will say that $x$ exhibits $\mathscr {F}$ as corepresented by $X$ if, for every object $Y \in \operatorname{\mathcal{C}}$, the composite map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \xrightarrow {\mathscr {F}} & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \mathscr {F}(X), \mathscr {F}(Y) ) \\ & \simeq & \operatorname{Fun}( \mathscr {F}(X), \mathscr {F}(Y) ) \\ & \xrightarrow {\operatorname{ev}_ x} & \mathscr {F}(Y) \end{eqnarray*}

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$; here the second map is the inverse of the homotopy equivalence $\operatorname{Fun}( \mathscr {F}(X), \mathscr {F}(Y) ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( X, Y)$ supplied by Remark 5.6.1.5.

We say that the functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$ if there exists a vertex $x \in \mathscr {F}(X)$ which exhibits $\mathscr {F}$ as corepresented by $X$. We say that the functor $\mathscr {F}$ is corepresentable if it is corepresentable by $X$, for some object $X \in \operatorname{\mathcal{C}}$.

Variant 5.7.6.2 (Representable Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and write $X^{\operatorname{op}}$ for the corresponding object of the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Given a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$, we say that a vertex $x \in \mathscr {F}(X^{\operatorname{op}} )$ exhibits $\mathscr {F}$ as represented by $X$ if it exhibits $\mathscr {F}$ as corepresented by the object $X^{\operatorname{op}}$, in the sense of Definition 5.7.6.1. We say that a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable by $X$ if it is corepresentable by $X^{\operatorname{op}}$, and that $\mathscr {F}$ is representable if it is representable by $X$ for some object $X \in \operatorname{\mathcal{C}}$.

Remark 5.7.6.3. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories and let $X \in \operatorname{\mathcal{C}}$ be an object, and let $x \in \mathscr {F}(X)$ be a vertex. The condition that $x$ exhibits $\mathscr {F}$ as corepresented by $X$ depends only on the connected component $[x] \in \pi _0( \mathscr {F}(X) )$.

Remark 5.7.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a morphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$, and let $x$ be a vertex of the Kan complex $\mathscr {F}(X)$. Then any two of the following conditions imply the third:

The vertex $x \in \mathscr {F}(X)$ exhibits the functor $\mathscr {F}$ as corepresented by $X$.

The vertex $\alpha (x) \in \mathscr {G}(X)$ exhibits the functor $\mathscr {G}$ as corepresented by $X$.

The natural transformation $\alpha $ is an isomorphism.

In particular, if $\mathscr {F}$ and $\mathscr {G}$ are isomorphic objects of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$, then $\mathscr {F}$ is corepresentable by $X$ if and only if $\mathscr {G}$ is corepresentable by $X$.

Remark 5.7.6.5. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Suppose we are given an object $Y \in \operatorname{\mathcal{D}}$ and a vertex $\eta \in \mathscr {F}( U(Y) )$. Then:

  • If $U$ is fully faithful and $\eta $ exhibits the functor $\mathscr {F}$ as corepresented by $U(Y)$, then it also exhibits the functor $\mathscr {F} \circ U$ as corepresented by $\eta $.

  • If $U$ is an equivalence of $\infty $-categories and $\eta $ exhibits the functor $\mathscr {F} \circ U$ as corepresented by $\eta $, then it also exhibits $\mathscr {F}$ as corepresented by $U(Y)$.

  • If $U$ is an equivalence of $\infty $-categories, then the functor $\mathscr {F}$ is corepresentable if and only if $\mathscr {F} \circ U$ is corepresentable.

Remark 5.7.6.6. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories, let $u: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, and let $x \in \mathscr {F}(X)$ be a vertex having image $y = \mathscr {F}(u)(x) \in \mathscr {F}(Y)$. Then any two of the following conditions imply the third:

  • The vertex $x$ exhibits the functor $\mathscr {F}$ as corepresented by $X$.

  • The vertex $y$ exhibits the functor $\mathscr {F}$ as corepresented by $Y$.

  • The morphism $u$ is an isomorphism.

Remark 5.7.6.7 (Uniqueness of the Corepresenting Object). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories which is corepresentable by an object $X \in \operatorname{\mathcal{C}}$. Let $Y$ be another object of $\operatorname{\mathcal{C}}$. Then $\mathscr {F}$ is corepresentable by $Y$ if and only if $Y$ is isomorphic to $X$. The “if” direction follows immediately from Remark 5.7.6.6. Conversely, suppose that $\mathscr {F}$ is corepresented by $Y$. Choose vertices $x \in \mathscr {F}(X)$ and $y \in \mathscr {F}(Y)$ which exhibit $\mathscr {F}$ as corepresented by $x$ and $y$, respectively. Since evaluation at $x$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \mathscr {F}(Y)$, we can choose a morphism $u: X \rightarrow Y$ such that $\mathscr {F}(u)(x)$ and $y$ belong to the same connected component of $\mathscr {F}(Y)$. Then $\mathscr {F}(u)(x)$ also exhibits $\mathscr {F}$ as corepresented by $Y$ (Remark 5.7.6.3), so that $u$ is an isomorphism in $\operatorname{\mathcal{C}}$ by virtue of Remark 5.7.6.6.

Remark 5.7.6.8. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories. Then the construction $Y \mapsto \pi _0( \mathscr {F}(Y) )$ determines a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets, which we will denote by $\pi _0( \mathscr {F} )$. Suppose that $X$ is an object of $\operatorname{\mathcal{C}}$ and $x \in \mathscr {F}(X)$ exhibits $\mathscr {F}$ as corepresented by $X$. Then, for every object $Y \in \operatorname{\mathcal{C}}$, evaluation on the connected component $[x] \in \pi _0( \mathscr {F}(X) )$ induces a bijection

\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) = \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) ) \rightarrow \pi _0( \mathscr {F}(Y) ). \]

It follows that the functor $\pi _0(\mathscr {F}): \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ is corepresentable by $X$, in the sense of classical category theory.

Warning 5.7.6.9. The converse of Remark 5.7.6.8 is false in general. For example, let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}\subset \operatorname{\mathcal{S}}$ be the functor given on objects by the formula $\mathscr {F}(Y) = \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$. Then $\pi _0(\mathscr {F} )$ is corepresentable by the object $X$ (when regarded as a functor from $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets), but the functor $\mathscr {F}$ is usually not corepresentable.

In spite of Warning 5.7.6.9, the corepresentability of a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ can be tested at the level of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. The caveat is that we must equip $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with the enrichment described in Construction 4.6.8.13.

Definition 5.7.6.10. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.5.10) and let $\operatorname{\mathcal{C}}$ be an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category containing an object $X$. We will say that an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is corepresentable by $X$ if there exists a vertex $x \in \mathscr {F}(X)$ such that, for every object $Y \in \operatorname{\mathcal{C}}$, the induced map

\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y ) \times \{ x\} \hookrightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y ) \times \mathscr {F}(X) \rightarrow \mathscr {F}(Y) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. In this case, we also say that $x$ exhibits $\mathscr {F}$ as corepresented by the object $X$. We say that the functor $\mathscr {F}$ is corepresentable if it is corepresentable by $X$ for some object $X \in \operatorname{\mathcal{C}}$.

We say that an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is representable by $X$ if it is corepresentable by the object $X^{\operatorname{op}} \in \operatorname{\mathcal{C}}^{\operatorname{op}}$, and that $\mathscr {F}$ is representable if it is representable by some object of $\operatorname{\mathcal{C}}$.

Remark 5.7.6.11. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories, and let $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}$ be the induced functor of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched homotopy categories (see Construction 4.6.8.13). Then:

  • A vertex $x \in \mathscr {F}(X)$ exhibits $\mathscr {F}$ as corepresented by an object $X \in \operatorname{\mathcal{C}}$ (in the sense of Definition 5.7.6.1) if and only if it exhibits $\mathrm{h} \mathit{\mathscr {F}}$ as corepresented by the object $X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (in the sense of Definition 5.7.6.10).

  • The functor $\mathscr {F}$ is corepresentable by an object $X \in \operatorname{\mathcal{C}}$ if and only if $\mathrm{h} \mathit{\mathscr {F}}$ is corepresentable by $X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

  • The functor $\mathscr {F}$ is corepresentable if and only if $\mathrm{h} \mathit{\mathscr {F}}$ is corepresentable as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor.

Remark 5.7.6.12. Let $\operatorname{\mathcal{C}}$ be an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category. Then an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is corepresentable by an object $X \in \operatorname{\mathcal{C}}$ if and only if it is isomorphic (as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor) to the functor

\[ \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad Y \mapsto \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y). \]

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. It follows from Remarks 5.7.6.4 and 5.7.6.7 that there is a unique function

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Isomorphism classes of corepresentable functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$} \} \ar [d] \\ \{ \textnormal{Isomorphism classes of objects of $\operatorname{\mathcal{C}}$} \} , } \]

which carries (the isomorphism class of) a corepresentable functor $\mathscr {F}$ to (the isomorphic class of) an object $X \in \operatorname{\mathcal{C}}$ which corepresents $\mathscr {F}$. Our main goal in this section is to show that, modulo set-theoretic considerations, this map is bijective.

Theorem 5.7.6.13. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. Then, for every object $X \in \operatorname{\mathcal{C}}$, there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by $X$. Moreover, the functor $\mathscr {F}$ is uniquely determined up to isomorphism.

Notation 5.7.6.14 (Corepresentable Functors). Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. For every object $X \in \operatorname{\mathcal{C}}$, Theorem 5.7.6.13 asserts that there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresented by $X$, which is uniquely determined up to isomorphism. To emphasize this uniqueness, we will typically denote the functor $\mathscr {F}$ by $h^{X}$ and refer to it as the functor corepresented by $X$. For every object $Y \in \operatorname{\mathcal{C}}$, we can apply the same argument to the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to obtain a functor represented by $Y$, which we will typically denote by $h_{Y}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ and refer to as the functor represented by $Y$. Note that Remark 5.7.6.12 supplies isomorphisms $h^{X}(Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \simeq h_{Y}(X)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, depending functorially on the pair $(X,Y) \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Remark 5.7.6.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every object $X \in \operatorname{\mathcal{C}}$ determines an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, \bullet ): \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y). \]

Theorem 5.7.6.13 asserts $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, \bullet )$ can be promoted, in an essentially unique way, to a functor of $\infty $-categories $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ (see Remark 5.7.6.12). Beware that this is a special feature of corepresentable functors. In general, an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathscr {F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ cannot be promoted to a functor of $\infty $-categories. Moreover, when such a promotion exists, it need not be unique.

Remark 5.7.6.16 (Functoriality). Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. We will see later that the corepresentable functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ and the representable functor $h_{Y}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ of Notation 5.7.6.14 depend functorially on the objects $X$ and $Y$, respectively. More precisely, the construction

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(\bullet , \bullet ): \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

can be promoted to a functor of $\infty $-categories $H: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ with the following properties:

  • For each object $X \in \operatorname{\mathcal{C}}$, the restriction $H|_{ \{ X\} \times \operatorname{\mathcal{C}}}$ is corepresentable by $X$.

  • For each object $Y \in \operatorname{\mathcal{C}}$, the restriction $H|_{ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \{ Y\} }$ is representable by $Y$.

See Proposition .

Unlike its classical counterpart, Theorem 5.7.6.13 is nontrivial: given an object $X$ of an $\infty $-category $\operatorname{\mathcal{C}}$, there is no immediately obvious candidate for a functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresented by $X$. However, the situation is better when $\operatorname{\mathcal{C}}$ arises from a simplicially enriched category.

Proposition 5.7.6.17. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let

\[ \mathscr {F}: \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}}. \]

denote the homotopy coherent nerve of the simplicial functor $Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$. Then the identity morphism $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } = \mathscr {F}(X)$ exhibits the functor $\mathscr {F}$ as corepresented by $X$, in the sense of Definition 5.7.6.1.

Proof. Fix an object $Y \in \operatorname{\mathcal{C}}$. We then have a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [r]^-{U} \ar [d]^{\theta }_{\sim } & \operatorname{Hom}_{ \operatorname{Kan}}( \mathscr {F}(X), \mathscr {F}(Y) )_{\bullet } \ar [d]^{\theta '}_{\sim } \ar [r]^-{\operatorname{ev}} & \mathscr {F}(Y) \\ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y) \ar [r]^-{V} & \operatorname{Hom}_{ \operatorname{\mathcal{S}}}( \mathscr {F}(X), \mathscr {F}(Y) ), & } \]

where the vertical maps are supplied by Construction 4.6.7.3 (applied in the simplicial categories $\operatorname{\mathcal{C}}$ and $\operatorname{Kan}$, respectively) and $\operatorname{ev}$ is given by evaluation at the vertex $\operatorname{id}_{X} \in \mathscr {F}(X)$. Let $\theta '^{-1}$ denote a homotopy inverse to $\theta '$ (which exists by virtue of Theorem 4.6.7.5). Proposition 5.7.6.17 asserts that the composition $\operatorname{ev}\circ \theta '^{-1} \circ V$ is a homotopy equivalence. Since $\theta $ is also a homotopy equivalence (Theorem 4.6.7.5), this is equivalent to the assertion that $\operatorname{ev}\circ U$ is a homotopy equivalence. This is clear: the composition $\operatorname{ev}\circ U$ is the identity map from the Kan complex $\mathscr {F}(Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ to itself. $\square$

Remark 5.7.6.18. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. The preceding proof shows that if $\operatorname{\mathcal{C}}$ satisfies the conclusion of Proposition 5.7.6.17, then it also satisfies the conclusion of Theorem 4.6.7.5: that is, the comparison map $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}(X,Y)$ is a homotopy equivalence for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$. Note however that we have already used Theorem 4.6.7.5 (applied to the simplicial category $\operatorname{Kan}$) implicitly to give the definition of a corepresentable functor in the $\infty $-categorical setting.

The rest of this section is devoted to the proof of Theorem 5.7.6.13. Fix a locally small $\infty $-category $\operatorname{\mathcal{C}}$ and an object $X \in \operatorname{\mathcal{C}}$. We can then use the dictionary of Corollary 5.7.0.6 to identify functors $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ with left fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having essentially small fibers. We will show that $\mathscr {F}$ is corepresentable by an object $X \in \operatorname{\mathcal{C}}$ if and only if the $\infty $-category $\operatorname{\mathcal{E}}$ has an initial object $\widetilde{X}$ satisfying $U( \widetilde{X} ) = X$ (Proposition 5.7.6.21). We will then show that this condition guarantees that $U$ is equivalent to the left fibration $U_0: \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ (Proposition 5.7.6.21). Combining these assertions, we see that a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$ if and only if it is a covariant transport representation for $U_0$, so that the existence and uniqueness assertions of Theorem 5.7.6.13 follow from Theorem 5.7.0.2.

Proposition 5.7.6.19. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories and let $\widetilde{X} \in \operatorname{\mathcal{D}}$ be an object having image $X = U( \widetilde{X} )$. The following conditions are equivalent:

$(1)$

There exists an equivalence $F: \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{D}}$ of left fibrations over $\operatorname{\mathcal{C}}$ satisfying $F( \operatorname{id}_ X) = \widetilde{X}$.

$(2)$

The object $\widetilde{X} \in \operatorname{\mathcal{D}}$ is initial (Definition 4.6.6.1).

$(3)$

For every left fibration $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, evaluation on the object $\widetilde{X}$ induces a trivial Kan fibration $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

$(4)$

For every left fibration $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, evaluation on the object $\widetilde{X}$ induces a bijection

\[ \pi _0( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ) \rightarrow \pi _0( \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}). \]

Proof. If $F: \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{D}}$ is an equivalence of left fibrations over $\operatorname{\mathcal{C}}$, then it is an equivalence of $\infty $-categories (Proposition 5.1.6.5). Since $\operatorname{id}_{X}: X \rightarrow X$ is initial when regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{X/}$ (Proposition 4.6.6.23), Corollary 4.6.6.21 guarantees that $\widetilde{X}$ is an initial object of $\operatorname{\mathcal{D}}$. This proves the implication $(1) \Rightarrow (2)$. The implication $(2) \Rightarrow (3)$ follows by combining Corollary 4.6.6.25 with Proposition 4.2.5.4, and the implication $(3) \Rightarrow (4)$ is immediate.

We will complete the proof by showing that $(4)$ implies $(1)$. Note that the object $\operatorname{id}_{X} \in \operatorname{\mathcal{C}}_{X/}$ satisfies condition $(1)$ and therefore also satisfies condition $(3)$. It follows that there exists a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [rr]^-{F} \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl]^{U} \\ & \operatorname{\mathcal{C}}& } \]

satisfying $F( \operatorname{id}_{X} ) = \widetilde{X}$. To complete the proof, it will suffice to show that if condition $(4)$ is satisfied, then $F$ is an equivalence of left fibrations over $\operatorname{\mathcal{C}}$. For every left fibration $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, we have a commutative diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ \pi _0( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ) \ar [rr]^-{\circ [F]} \ar [dr] & & \pi _0( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}_{X/}, \operatorname{\mathcal{E}}) ) \ar [dl] \\ & \pi _0( \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}) & , } \]

where the vertical maps are given by evaluation on the objects $\widetilde{X} \in \operatorname{\mathcal{D}}$ and $\operatorname{id}_{X} \in \operatorname{\mathcal{C}}_{X/}$, and are therefore bijective. It follows that the horizontal map is also bijective. $\square$

Corollary 5.7.6.20. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix { \operatorname{\mathcal{D}}\ar [rr]^{F} \ar [dr]^{U} & & \operatorname{\mathcal{E}}\ar [dl]_{V} \\ & \operatorname{\mathcal{C}}& } \]

where $U$ and $V$ are left fibrations. Let $\widetilde{X} \in \operatorname{\mathcal{D}}$ be an initial object. Then $F$ is an equivalence of $\infty $-categories if and only if $F(\widetilde{X})$ is an initial object of $\operatorname{\mathcal{E}}$.

Proof. If $F$ is an equivalence of $\infty $-categories, then it carries initial objects to initial objects by virtue of Corollary 4.6.6.21. Conversely, suppose that $F( \widetilde{X} )$ is an initial object of $\operatorname{\mathcal{E}}$; we wish to show that $F$ is an equivalence of $\infty $-categories. Set $X = U( \widetilde{X} )$. Applying Proposition 5.7.6.19, we deduce that there is a functor $G \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{D}})$ such that $(G \circ F)( \widetilde{X} )$ is isomorphic to $\widetilde{X}$ as an object of the $\infty $-category $\operatorname{\mathcal{D}}_{X} = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. Applying Proposition 5.7.6.19 again, we deduce that $G \circ F$ is isomorphic to $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$; in particular, $F$ is a right homotopy inverse to $G$. Since $G$ carries $F( \widetilde{X} )$ to an initial object of $\operatorname{\mathcal{D}}$, we can apply the same argument (with the roles of $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ reversed) to show that $G$ has a left homotopy inverse. It follows that $G$ is an equivalence of $\infty $-categories, so that $F$ is also an equivalence of $\infty $-categories. $\square$

Proposition 5.7.6.21. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories having essentially small fibers, and let $X \in \operatorname{\mathcal{C}}$ be an object. Then:

$(1)$

Let $\widetilde{X} \in \operatorname{\mathcal{D}}$ be an object satisfying $U( \widetilde{X} ) = X$. Then $\widetilde{X}$ is an initial object of $\operatorname{\mathcal{D}}$ if and only if, for every object $Y \in \operatorname{\mathcal{C}}$, the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\theta } \operatorname{Fun}( \operatorname{\mathcal{D}}_{X}, \operatorname{\mathcal{D}}_{Y} ) \xrightarrow { \operatorname{ev}_{\widetilde{X}} } \operatorname{\mathcal{D}}_{Y} \]

is a homotopy equivalence, where $\theta $ is given by parametrized covariant transport (see Definition 5.2.8.1).

$(2)$

Let $\operatorname{hTr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ be the homotopy transport representation of $U$, which we regard as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor (Variant 5.2.8.12). Then $\widetilde{X}$ is an initial object of $\operatorname{\mathcal{D}}$ if and only if it exhibits $\operatorname{hTr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}$ as corepresented by $X$, in the sense of Definition 5.7.6.10.

$(3)$

The homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}$ is corepresentable by the object $X$ if and only if there exists an initial object $\widetilde{X} \in \operatorname{\mathcal{D}}$ satisfying $U( \widetilde{X} ) = X$.

$(4)$

Let $\operatorname{Tr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $U$. Then $\operatorname{Tr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}$ is corepresentable by the object $X$ if and only if there exists an initial object $\widetilde{X} \in \operatorname{\mathcal{D}}$ satisfying $U( \widetilde{X} ) = X$.

Proof. Let $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ be the oriented fiber product of Definition 4.6.4.1, and let us regard $\operatorname{id}_{X}$ as an initial object of $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ (Proposition 4.6.6.23). Using Proposition 5.7.6.19, we can choose a functor of $\infty $-categories $F: \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $F( \operatorname{id}_ X ) = \widetilde{X}$ which fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [rr]^-{F} \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl]^{U} \\ & \operatorname{\mathcal{C}}. & } \]

Using Proposition 5.7.6.19, we see that $\widetilde{X}$ is an initial object of $\operatorname{\mathcal{D}}$ if and only if $F$ is an equivalence of left fibrations over $\operatorname{\mathcal{C}}$. By virtue of Corollary 5.1.6.15, this is equivalent to the requirement that for each object $Y \in \operatorname{\mathcal{C}}$, the functor $F$ restricts to a homotopy equivalence of Kan complexes

\[ F_{Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{\mathcal{D}}_{Y} \]

Assertion $(1)$ follows from the observation that $F_{Y}$ is homotopic to the composition of the parametrized covariant transport morphism $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}_{X}, \operatorname{\mathcal{D}}_{Y} )$ with the evaluation map $\operatorname{ev}_{ \widetilde{X} }: \operatorname{Fun}( \operatorname{\mathcal{D}}_{X}, \operatorname{\mathcal{D}}_{Y} ) \rightarrow \operatorname{\mathcal{D}}_{Y}$ (see Remark 5.2.8.5 and Proposition 5.2.8.7). The implication $(1) \Rightarrow (2)$ follows from Remark 5.7.5.8, the implication $(2) \Rightarrow (3)$ is immediate, and the implication $(3) \Rightarrow (4)$ follows from Remark 5.7.6.11. $\square$

Proof of Theorem 5.7.6.13. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We wish to show that there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by $X$, and that $\mathscr {F}$ is uniquely determined up to isomorphism (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$. By virtue of Proposition 5.7.6.21 and Corollary 5.7.0.6, this is equivalent to the assertion that there exists a left fibration $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ together with an initial object $\widetilde{X} \in \operatorname{\mathcal{D}}$ satisfying $U( \widetilde{X}) = X$, and that the left fibration $U$ is uniquely determined up to equivalence (in the sense of Definition 5.1.6.1). To prove existence, we can take $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}_{X/}$ and $\widetilde{X}$ to be the identity morphism $\operatorname{id}_{X}$ (Proposition 4.6.6.23). The uniqueness assertion follows from Proposition 5.7.6.19. $\square$