nLab
(infinity,1)presheaf
Context
$(\infty,1)$Category theory
(∞,1)category theory
Background
Basic concepts
Universal constructions
Local presentation
Theorems
Models
$(\infty,1)$Topos Theory
(∞,1)topos theory
Background
Definitions

elementary (∞,1)topos

(∞,1)site

reflective sub(∞,1)category

(∞,1)category of (∞,1)sheaves

(∞,1)topos

(n,1)topos, ntopos

(∞,1)quasitopos

(∞,2)topos

(∞,n)topos
Characterization
Morphisms
Extra stuff, structure and property

hypercomplete (∞,1)topos

over(∞,1)topos

nlocalic (∞,1)topos

locally nconnected (n,1)topos

structured (∞,1)topos

locally ∞connected (∞,1)topos, ∞connected (∞,1)topos

local (∞,1)topos

cohesive (∞,1)topos
Models
Constructions
structures in a cohesive (∞,1)topos
Contents
Definition
Write $(\infty,0)Cat$ for the category ∞Grpd of $\infty$groupoids regarded as an (∞,1)category.
Let $S$ be a simplicial set (which in particular may be a quasicategory).
An $(\infty,1)$presheaf on $S$ is an (∞,1)functor
$F : S^{op} \to (\infty,0)Cat
\,.$
The (∞,1)category of $(\infty,1)$presheaves is the corresponding (∞,1)category of (∞,1)functors
$PSh(S) := Fun(S^{op}, (\infty,0)Cat)
\,.$
References
Section 5.1 of