(2,1)-quasitopos?
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 $\chi$ 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 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 $S$, 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?.
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]
Last revised on June 27, 2022 at 10:06:58. See the history of this page for a list of all contributions to it.