(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
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 .
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. 220.127.116.11) and as (Lurie, def. 4.3.3; def. 4.3.13).
Derived étale geometry
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.
This is (Lurie, def. 4.3.1).
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).
In its presentation as a model site the étale -site is given in definition 18.104.22.168 of
A discussion in the context of structured (∞,1)-toposes is