nLab cd-structure

Idea

Certain Grothendieck topologies on a category with an initial object can be presented by a very simple structure called a cd-structure.

A Grothendieck topology is “completely decomposable” if it is generated by covering families consisting of pairs of maps which are part of a distinguished commuting square. The collection of these distinguished squares is then called a “cd-structure”.

The descent-condition for simplicial presheaves (their $\infty$-stack-condition) over completely decomposable topologies may be expressed by a Brown-Gersten property, hence a generalized Mayer-Vietoris property, which essentially says that a simplicial presheaf takes distinguished squares to homotopy pullback-squares. Typically this is much easier to check than the generic (homotopy-)descent condition.

Definition

Definition

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.

Properties

Any cd-structure gives rise in a canonical way to a Grothendieck topology on $C$.

Definition

Let $\chi$ be a cd-structure on $C$. The associated Grothendieck topology $\tau_\chi$ on $C$ is the coarsest topology such that:

• the empty sieve covers $\emptyset_C$, the initial object of $C$;
• $\{ b \to d, c \to d \}$ generates a covering sieve for all $\chi$-distinguished squares
  $$\begin{matrix} a& \to & b \\ \downarrow& &\, \downarrow p\\ c &\underset{j}{\to} & d \end{matrix}$$

Definition

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 $C$.

Definition

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

$$\begin{matrix} a& \to & b \\ \downarrow& &\, \downarrow\\ a \times_c a &\to & b \times_d b \end{matrix}$$

belongs to $\chi$.

Proposition

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 2010, Lem. 2.9 and Prop. 2.15. For general (∞,1)-presheaves, this is a rephrasing of Voevodsky 2010, Cor. 5.10.

Examples

There are various interesting cd-structures on the category of schemes over a base $S$, which give rise to Grothendieck topologies like

• the Zariski topology. Here a square is distinguished iff the cospan part formed by open embeddings, which together form an open cover, and the square is a pullback square.

• the Nisnevich topology;

• the cdh topology?.

See also

• Grothendieck topology
• Nisnevich topology
• cdh topology?

References

• Brad Drew, Descente: Nisnevich et cdh, Groupe de travail at Université Paris 13 (Spring 2010) [pdf]

• Vladimir Voevodsky, Section 2 of: Homotopy theory of simplicial presheaves in completely decomposable topologies, Journal of Pure and Applied Algebra 214 8 (2010) 1384-1398 [arXiv:0805.4578, doi:10.1016/j.jpaa.2009.11.004]

• Vladimir Voevodsky, Unstable motivic homotopy categories in Nisnevich and cdh-topologies, Journal of Pure and Applied Algebra 214 8 (2010) 1399-1406 [arXiv:0805.4576]