nLab
(infinity,1)-category of (infinity,1)-presheaves

Definition

For ∞Grpd the (∞,1)-category of ∞-groupoids, and for $S$ a (∞,1)-category (or in fact any simplicial set), an $(\mathrm{\infty }\mathrm{,1})$-presheaf on $S$ is an $(\mathrm{\infty }\mathrm{,1})$-functor

$$F:S^{\mathrm{op}}\to \mathrm{\infty }\mathrm{Grpd}.$$F : S^{op} \to \infty Grpd \,.

The $(\mathrm{\infty }\mathrm{,1})$-category of $(\mathrm{\infty }\mathrm{,1})$-presheaves is the (∞,1)-category of (∞,1)-functors

$${\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(S):=\mathrm{Func}(S^{\mathrm{op}},\mathrm{\infty }\mathrm{Grpd}).$$PSh_{(\infty,1)}(S) := Func(S^{op}, \infinity Grpd) \,.

Models

A model for such $(\mathrm{\infty }\mathrm{,1})$-presheaf categories is the model structure on simplicial presheaves. See also the discussion at models for ∞-stack (∞,1)-toposes.

Related entries

see also

• presentable (∞,1)-category

• (∞,1)-category of (∞,1)-sheaves

References

section 5.1 of

• Jacob Lurie, Higher Topos Theory