nLab separated (2,1)-presheaf

Definition

Definition

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

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

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

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

If this morphism is even an equivalence of categories, then $X$ 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 $(2,1)$-presheaf .

Examples

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

$$\bar W G \to G Bund$$

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

Related concepts

• separated presheaf

• separated $(2,1)$-presheaf

• separated (∞,1)-presheaf