nLab
separated (2,1)-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 (2,1)-presheaf/prestack over a (2,1)-site CC is a (2,1)-presheaf X:C opGrpdX : C^{op} \to Grpd such that covering families {U iU}\{U_i \to U\} in CC the descent morphism

X(U)PSh (2,1)(S({U i}),X) X(U) \to PSh_{(2,1)}(S(\{U_i\}), X)

is a full and faithful functor and hence exhibits a full subcategory.

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

If this morphism is even an equivalence of categories, then XX is even a (2,1)-sheaf/stack.

Remark on terminology

The term prestack is used in two different ways in the literature: some authors use it synonymously with just (2,1)-presheaf, others with separated (2,1)(2,1)-presheaf .

Examples

Let Mfd be the site of topological manifolds. Let GG be a topological group and W¯G\bar W G the (2,1)-presheaf on MfdMfd represented by the nerve of the delooping-groupoid (see simplicial group for te notation). Let GBundG Bund be the (2,1)-sheaf of all GG-principal bundles. This is the (2,1)-sheafification of W¯G\bar W G. The canonical morphism

W¯GGBund \bar W G \to G Bund

includes over each UMfdU \in Mfd the single object of (W¯G)(U)(\bar W G)(U) as the trivial GG-principal bundle. Its automorphisms are given by continuous functions C(U,G)C(U,G). This is the same on both sides, hence (W¯G)(U)GBund(U)(\bar W G)(U) \to G Bund(U) is a full and faithful functor and W¯G\bar W G is a separated (2,1)(2,1)-presheaf.

Created on January 26, 2011 16:42:51 by Urs Schreiber (89.204.153.99)