Certain Grothendieck topologies on a category with an initial object can be presented by a very simple structure called a cd-structure. For such sites, the condition of descent for sheaves can be checked via a Mayer-Vietoris-like property.
A cd-structure $\chi$ on a category $C$ with an initial object is a class of commutative squares which is stable by isomorphism. We will call its elements $\chi$-distinguished squares.
Any cd-structure gives rise in a canonical way to a Grothendieck topology on $C$.
Let $\chi$ be a cd-structure on $C$. The associated Grothendieck topology $\tau_\chi$ on $C$ is the coarsest topology such that:
A cd-structure $\chi$ is called complete if every morphism whose target is an initial object of $C$ is an isomorphism, and if it is stable by base change along arbitrary morphisms of $C$.
A cd-structure $\chi$ is called regular if (i) each $\chi$-distinguished square $Q$ is cartesian, (ii) the lower horizontal morphism $j : c \to d$ is a monomorphism, and (iii) for each $Q \in \chi$, the induced commutative square
belongs to $\chi$.
Let $\chi$ be a complete cd-structure. If $F$ is an (∞,1)-presheaf on $C$ that maps initial objects of $C$ to terminal objects, and sends $\chi$-distinguished squares to cartesian squares, then $F$ is a $\tau_\chi$-(∞,1)-sheaf.
If $\chi$ is further regular, then the converse is also true.
For presheaves of sets, this is Voevodsky, Lem. 2.9 and Prop. 2.15. For general (∞,1)-presheaves, this is a rephrasing of Voevodsky, Cor. 5.10.
There are various interesting cd-structures on the category of schemes over a base $S$, which give rise to Grothendieck topologies like the Zariski, Nisnevich and cdh? topologies.
Brad Drew, Descente : Nisnevich et cdh, Groupe de travail at Université Paris 13, Spring 2010, pdf.
Vladimir Voevodsky, Homotopy theory of simplicial presheaves in completely decomposable topologies, 2000, K-theory archive, arXiv:0805.4578.
Nisnevich and cdh-topologies_, 2000, arXiv:0805.4576.
Last revised on February 24, 2015 at 22:26:41. See the history of this page for a list of all contributions to it.