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equivalences in/of $(\infty,1)$-categories
The statement of the Yoneda lemma has a straightforward generalization from categories to (∞,1)-categories.
For $C$ an (∞,1)-category and $PSh(C)$ its (∞,1)-category of (∞,1)-presheaves, the $(\infty,1)$-Yoneda embedding is the (∞,1)-functor
given by $y(X) : U \mapsto C(U,X)$.
$(\infty,1)$-Yoneda embedding
Let $C$ be an (∞,1)-category and $PSh(C) := Func(C^\op, \infty Grpd)$ be the corresponding (∞,1)-category of (∞,1)-presheaves. Then the canonical (∞,1)-functor
$(\infty,1)$-Yoneda theorem
For $C$ a small $(\infty,1)$-category and $F : C^{op} \to \infty Grpd$ an $(\infty,1)$-functor, the composite
is equivalent to $F$.
This appears as HTT Lemma 5.5.2.1.
The statement is a direct consequence of the sSet-enriched Yoneda lemma by using the fact that the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C)$ is modeled by the enriched functor category $[C^{op}, sSet]_{proj}$ with $C$ regarded as a simplicially enriched category and using the global model structure on simplicial presheaves.
The $(\infty,1)$-Yoneda embedding $y : C \to PSh(C)$ preserves all (∞,1)-limits that exist in $C$.
This appears as HTT, prop. 5.1.3.2.
For $C$ an (∞,1)-site and $\mathcal{X}$ an (∞,1)-topos, (∞,1)-geometric morphisms $(f^* \dashv f_*) Sh(C) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{X}$ from the (∞,1)-sheaf (∞,1)-topos $Sh(C)$ to $\mathcal{X}$ correspond to the local (∞,1)-functors $f^* : C \to \mathcal{X}$, those that
are left exact (∞,1)-functors;
send covering families $\{U_i \to X\}$ in $\mathcal{G}$ to effective epimorphism
More preseicely, the (∞,1)-functor
given by precomposition of inverse image functors by ∞-stackification and by the (∞,1)-Yoneda embedding is a full and faithful (∞,1)-functor and its essential image is spanned by these local morphisms.
This appears as (HTT, prop. 6.2.3.20).
Published statements appear in
as indicated above.
See also the discussion on MathOverflow.