nLab separated (infinity,1)-presheaf

Context

$(\infty,1)$-Topos Theory

(∞,1)-topos theory

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

Definition

A separated (∞,1)-presheaf over an (∞,1)-site $C$ is a (∞,1)-presheaf $X : C^{op} \to$ ∞Grpd such that covering families $\{U_i \to U\}$ in $C$ the descent comparison morphism

$X(U) \to PSh_{(\infty,1)}(S(\{U_i\}), X)$

is a full and faithful (∞,1)-functor and hence exhibits a full sub-(∞,1)-category.

(Here $S(\{U_i\})$ denotes the sieve associated to the cover).

More generally, $X$ is $k$-separated for $k \in \mathbb{N}$ if the descent morphism is a $(k-2)$-truncated morphism.

Notice that this means that a 0-separated $(\infty,1)$-presheaf is one whose descent morphisms are equivalences, hence those which are (∞,1)-sheaves.

Revised on September 11, 2011 17:55:08 by Urs Schreiber (82.113.99.24)