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8.3 Presheaf $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category. It is then very rare for $\operatorname{\mathcal{C}}$ to admit small colimits: this is possible only if $\operatorname{\mathcal{C}}$ is (equivalent to the nerve of) a partially ordered set (Proposition ). However, it is always possible to embed $\operatorname{\mathcal{C}}$ into a larger $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$. For example, we can take $\widehat{\operatorname{\mathcal{C}}}$ to be the $\infty $-category of functors $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. Theorem 8.2.5.5 implies that the covariant Yoneda embedding

\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto h_{Y} \]

is fully faithful. Moreover, the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ admits small colimits, which can be computed levelwise: this is a special case of Proposition 7.1.6.1, since the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ admits small colimits (Corollary 7.4.5.6).

Our goal in this section is to show that the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is universal among functors from $\operatorname{\mathcal{C}}$ to $\infty $-categories which admit small colimits. More precisely, suppose that $\operatorname{\mathcal{D}}$ is an $\infty $-category which admits small colimits, and let $\operatorname{Fun}'( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}})$ spanned by those functors which preserve small colimits. In §8.3.3, we will show that precomposition with $h_{\bullet }$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}'( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]

(Theorem 8.3.3.1). Stated more informally, the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ can be obtained from $\operatorname{\mathcal{C}}$ by freely adjoining colimits of small diagrams.

Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is small and $\operatorname{\mathcal{D}}$ admits small colimits. Theorem 8.3.3.1 implies that $f$ admits an essentially unique extension to a colimit-preserving functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$. This extension admits other useful characterizations:

$(a)$

It is a left Kan extension of $f$ along the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (see Example 8.3.3.11).

$(b)$

If $\operatorname{\mathcal{D}}$ is locally small, we will prove in §8.3.4 that $F$ is left adjoint to the composite functor

\[ \operatorname{\mathcal{D}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow { \circ f^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \]

(Proposition 8.3.4.1).

To prove Theorem 8.3.3.1, the main step is to show that property $(a)$ is satisfied by the identity functor $\operatorname{id}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. This is equivalent to the assertion that the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is a dense functor: that is, it exhibits each object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ as a colimit of the diagram

\[ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \]

(see Definition 8.3.1.15). In §8.3.1, we discuss dense functors in general and provide a concrete criterion which can be used to show that a functor is dense (Proposition 8.3.1.23). In §8.3.2, we apply this criterion to establish the density of the Yoneda embedding $h_{\bullet }$ (Theorem 8.3.2.1).

Let us isolate another important feature of the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. For every pair of objects $X \in \operatorname{\mathcal{C}}$ and $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, the $\infty $-categorical analogue of Yoneda's lemma supplies a homotopy equivalence

\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})}( h_{X}, \mathscr {F} ) \xrightarrow {\sim } \mathscr {F}(X) \]

(Proposition 8.2.1.3). It follows that $h_{X}$ is an atomic object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$: that is, it corepresents a functor which preserves small colimits (Definition 8.3.5.1). The Yoneda embedding is essentially characterized by this property, together with the fact that it is dense and fully faithful. More precisely, suppose that we are given a functor of $\infty $-categories $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{C}}$ is small and $\operatorname{\mathcal{D}}$ admits small colimits. Theorem 8.3.3.1 then guarantees that $f$ admits an essentially unique colimit-preserving extension $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. In §8.3.5, we show that $F$ is an equivalence of $\infty $-categories if and only if $\operatorname{\mathcal{D}}$ is locally small and the functor $f$ is dense, fully faithful, and carries each object of $\operatorname{\mathcal{C}}$ to an atomic object of $\operatorname{\mathcal{D}}$ (Proposition 8.3.5.6).

Structure

  • Subsection 8.3.1: Dense Functors
  • Subsection 8.3.2: Density of Yoneda Embeddings
  • Subsection 8.3.3: The Universal Property of Presheaf $\infty $-Categories
  • Subsection 8.3.4: Example: Extensions as Adjoints
  • Subsection 8.3.5: Characterization of Yoneda Embeddings