nLab
(2,1)-sheaf

Context

Locality and descent

2-Category theory

(,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

Idea

A (2,1)(2,1)-sheaf is a sheaf with values in groupoids. This is traditionally called a stack.

Definition

Let CC be a (2,1)-site. Write Grpd for the (2,1)-category of groupoids, functors and natural isomorphisms.

A (2,1)(2,1)-sheaf on CC is equivalently

The (2,1)(2,1)-category of (2,1)(2,1)-sheaves

The (2,1)-category of a (2,1)(2,1)-sheaves on a (2,1)-site forms a (2,1)-topos.

There are model category presentations of this (2,1)(2,1)-topos. See model structure for (2,1)-sheaves.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoidh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-\infty-groupoid

Revised on April 25, 2013 22:00:22 by Urs Schreiber (82.169.65.155)