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 prsheaves 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

$$U\stackrel{{\mathrm{Id}}_U{\times }_X{\mathrm{Id}}_U}{\to }U{\times }_{X}U$$U \stackrel{Id_U \times_X 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)