nLab
(infinity,1)-presheaf

Definition

Write $(\mathrm{\infty },0)\mathrm{Cat}$ for the category ∞Grpd of $\mathrm{\infty }$-groupoids regarded as an (∞,1)-category.

Let $S$ be a simplicial set (which in particular may be a quasi-category).

An $(\mathrm{\infty },1)$-presheaf on $S$ is an (∞,1)-functor

$$F:S^{\mathrm{op}}\to (\mathrm{\infty },0)\mathrm{Cat}.$$F : S^{op} \to (\infty,0)Cat \,.

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

$$\mathrm{PSh}(S):=\mathrm{Fun}(S^{\mathrm{op}},(\mathrm{\infty },0)\mathrm{Cat}).$$PSh(S) := Fun(S^{op}, (\infty,0)Cat) \,.

Remarks

• $(\mathrm{\infty },1)$-presheaves are the basic building block for the definition of (∞,1)-categories of (∞,1)-sheaves.

Related concepts

• presheaf

• (2,1)-presheaf

• $(\mathrm{\infty },1)$-presheaf

• (∞,n)-presheaf

References

Section 5.1 of

• Jacob Lurie, Higher Topos Theory .