# nLab (2,1)-sheaf

Contents

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

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

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos

# Contents

## Idea

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

## Definition

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

A $(2,1)$-sheaf on $C$ is equivalently

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

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

There are model category presentations of this $(2,1)$-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)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-$\infty$-groupoid

Last revised on April 25, 2013 at 22:00:22. See the history of this page for a list of all contributions to it.