nLab
cd-structure

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

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.

Definition

Definition

A cd-structure χ\chi on a category CC with an initial object is a class of commutative squares which is stable by isomorphism. We will call its elements χ\chi-distinguished squares.

Properties

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

Definition

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

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

A cd-structure χ\chi is called complete if every morphism whose target is an initial object of CC is an isomorphism, and if it is stable by base change along arbitrary morphisms of CC.

Definition

A cd-structure χ\chi is called regular if (i) each χ\chi-distinguished square QQ is cartesian, (ii) the lower horizontal morphism j:cdj : c \to d is a monomorphism, and (iii) for each QχQ \in \chi, the induced commutative square

a b a× ca b× db \begin{matrix} a& \to & b \\ \downarrow& &\, \downarrow\\ a \times_c a &\to & b \times_d b \end{matrix}

belongs to χ\chi.

Proposition

Let χ\chi be a complete cd-structure. If FF is an (∞,1)-presheaf on CC that maps initial objects of CC to terminal objects, and sends χ\chi-distinguished squares to cartesian squares, then FF is a τ χ\tau_\chi-(∞,1)-sheaf.

If χ\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.

Examples

There are various interesting cd-structures on the category of schemes over a base SS, which give rise to Grothendieck topologies like the Zariski, Nisnevich and cdh? topologies.

See also

References

Last revised on February 24, 2015 at 22:26:41. See the history of this page for a list of all contributions to it.