nLab separated (infinity,1)-presheaf

Definition

Definition

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

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

is a full and faithful (∞,1)-functor and hence exhibits a full sub-(∞,1)-category.

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

More generally, $X$ is $k$-separated for $k \in \mathbb{N}$ if the descent morphism is a $(k-2)$-truncated morphism.

Notice that this means that a 0-separated $(\infty,1)$-presheaf is one whose descent morphisms are equivalences, hence those which are (∞,1)-sheaves.

Related concepts

• separated presheaf

• separated (2,1)-presheaf

• separated (∞,1)-presheaf