(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
The étale (∞,1)-site is an (∞,1)-site whose (∞,1)-topos encodes the derived geometry version of the geometry encoded by the topos over the étale site.
Its underlying (∞,1)-category is the opposite (∞,1)-category of commutative simplicial algebras over a commutative ring , whose covering families are essentially those which under decategorification become coverings in the étale site of ordinary -algebras.
Let be a commutative ring. Let be the Lawvere theory of commutative associative algebras over .
Let be the sSet-enriched category of simplicial commutative associative k-algebras equipped with the standard model structure on simplicial T-algebras. Write for the -category presentable (∞,1)-category. Then we have an equivalence of (∞,1)-categories
This is a special case of the general statement discussed at (∞,1)-algebraic theory. See also (Lurie, remark 4.1.2).
For we write for the corresponding object in and conversely for we write for the corresponding object in .
So and , by definition of notation.
Notice from the discussion at model structure on simplicial algebras the homotopy group functor
A morphism in is an étale morphism if
The underlying morphism is an étale morphism of schemes;
for each the canonical morphism
is an isomorphism.
The étale -site is the (∞,1)-site whose underlying -category is the opposite (∞,1)-category and whose covering famlies are those collections of morphisms such that
every is an étale morphism
there is a finite subset such that the underlying decategorified family is a covering family in the 1-étale site.
This appears as (ToënVezzosi, def. 2.2.2.12) and as (Lurie, def. 4.3.3; def. 4.3.13).
The following definition and theorem show how the étale -site arises naturally from the étale 1-site, and naturally encodes the derived geometry induced by the étale site.
(étale pregeometry)
Let be the 1-étale site regarded as a pregeometry (for structured (∞,1)-toposes) as follows.
the underlying (∞,1)-category is the 1-category
which is the full subcategory of on those objects for which there exists an étale morphism from the polynomial algebra in generators for some ;
the admissible morphisms in the pregeometry are the étale morphisms;
a collection of admissible morphisms is a covering family if it is so as a family of morphisms in the étale site.
This is (Lurie, def. 4.3.1).
(étale geometry)
Let be the geometry (for structured (∞,1)-toposes) given by
the underlying (∞,1)-site is the étale -site;
the admissible morphisms are the étale morphisms.
This is (Lurie, def. 4.3.13).
The geometry generated by the étale pregeometry is the étale geometry .
This is (Lurie, prop. 4.3.15).
étale -site
In its presentation as a model site the étale -site is given in definition 2.2.2.12 of
.
A discussion in the context of structured (∞,1)-toposes is
See also
Last revised on May 28, 2022 at 15:18:26. See the history of this page for a list of all contributions to it.