nLab
separated (infinity,1)-presheaf

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

Definition

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

X(U)PSh (,1)(S({U i}),X) 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})S(\{U_i\}) denotes the sieve associated to the cover).

More generally, XX is kk-separated for kk \in \mathbb{N} if the descent morphism is a (k2)(k-2)-truncated morphism.

Notice that this means that a 0-separated (,1)(\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)