Contents

topos theory

# 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 $\chi$ 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, 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 $S$, which give rise to Grothendieck topologies like the Zariski, Nisnevich and cdh? topologies.