nLab Verdier site

Idea

A Verdier site (named after Jean-Louis Verdier) is a site with an extra condition on the coverage that ensures that one can refine every hypercover in the context of simplicial presheaves on that site in a particularly nice.

For a discussion of the use of Verdier sites see descent.

Definition

A Verdier site is a site such that for every morphism $U \to X$ which is part of a covering sieve on $X$, also

$$U \stackrel{(Id_U,Id_U)}{\to} U \times_X U$$

is in a covering sieve of $U \times_X U$.

References

section 9 of

• Daniel Dugger, Sharon Hollander, Daniel Isaksen, Hypercovers and simplicial presheaves (arXiv)