nLab (2,1)-sheaf

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

• a 2-functor $C^{op} \to Grpd$ that satisfies descent;

• a 2-sheaf with values in groupoids;

• a 1-truncated (∞,1)-sheaf on $C$.

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.

Related concepts

• sheaf

• 2-sheaf / $(2,1)$-sheaf / stack

• (∞,1)-sheaf / ∞-stack

• (∞,n)-sheaf

• descent

  • cover

  • cohomological descent

  • descent morphism

  • monadic descent,

    • Sweedler coring

    • higher monadic descent

    • descent in noncommutative algebraic geometry

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid