nLab
Yoneda lemma for (infinity,1)-categories

Idea

The statement of the Yoneda lemma has a straightforward generalization from categories to (∞,1)-categories.

Following these answers on MathOverflow, the full $(\mathrm{\infty }\mathrm{,1})$-statement obtained this way holds.

One aspect of the generalization of the standard Yoneda lemma, the fact that the Yoneda embedding is a full and faithful functor, is known in the published literature to generalize.

Details

Theorem

$(\mathrm{\infty }\mathrm{,1})$-Yoneda embedding

Let $C$ be an (∞,1)-category and $\mathrm{PSh}(C):=\mathrm{Func}(C^{op},\mathrm{\infty }\mathrm{Grpd})$ be the corresponding (∞,1)-category of (∞,1)-presheaves. Then the canonical (∞,1)-functor

$$Y:C\to \mathrm{PSh}(C)$$Y : C \to PSh(C)

is a full and faithful (∞,1)-functor.

Proof

This is proposition 5.1.3.1 in

• Jacob Lurie, Higher Topos Theory