nLab cd-structure



Topos Theory

topos theory



Internal Logic

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In higher category theory


(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



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.



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.


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


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}

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.


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.


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


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

See also


Last revised on June 27, 2022 at 10:06:58. See the history of this page for a list of all contributions to it.