(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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 $sCAlg_k^{op}$ of commutative simplicial algebras over a commutative ring $k$, whose covering families are essentially those which under decategorification become coverings in the étale site of ordinary $k$-algebras.
Let $k$ be a commutative ring. Let $T$ be the Lawvere theory of commutative associative algebras over $k$.
Let
be the (∞,1)-category of ∞-algebras over $T$ regarded as an (∞,1)-algebraic theory.
Let $sCAlg_k = (T Alg)^{\Delta^{op}}$ be the sSet-enriched category of simplicial commutative associative k-algebras equipped with the standard model structure on simplicial T-algebras. Write $sCAlg_k^\circ$ for the $(\infty,1)$-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 $X \in \infty CAlg_k^{op}$ we write $\mathcal{O}(X)$ for the corresponding object in $\infty CAlg_k$ and conversely for $A \in \infty CAlg_k$ we write $Spec A$ for the corresponding object in $\infty CAlg_k^{op}$.
So $Spec \mathcal{O} X = X$ and $\mathcal{O} Spec A = A$, by definition of notation.
Notice from the discussion at model structure on simplicial algebras the homotopy group functor
A morphism $Spec A \to Spec B$ in $\infty CAlg_k^{op}$ is an étale morphism if
The underlying morphism $Spec \pi_0(A) \to Spec \pi_0(B)$ is an étale morphism of schemes;
for each $i \in \mathbb{N}$ the canonical morphism
is an isomorphism.
The étale $(\infty,1)$-site is the (∞,1)-site whose underlying $(\infty,1)$-category is the opposite (∞,1)-category $\infty CAlg_k^{op}$ and whose covering famlies $\{Spec A_i \to Spec B\}_{i \in I}$ are those collections of morphisms such that
every $Spec A_i \to Spec B$ is an étale morphism
there is a finite subset $J \subset I$ such that the underlying decategorified family $\{Spec \pi_0(A_j) \to Spec \pi_0(B)\}_{j \in J}$ 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 $(\infty,1)$-site arises naturally from the étale 1-site, and naturally encodes the derived geometry induced by the étale site.
(étale pregeometry)
Let $\mathcal{T}_{et}$ 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 $CAlg_k$ on those objects $A \in CAlg_k$ for which there exists an étale morphism $k[x^1, \cdots, x^n] \to A$ from the polynomial algebra in $n$ generators for some $n \in \mathbb{N}$;
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 $\mathcal{G}_{et}$ be the geometry (for structured (∞,1)-toposes) given by
the underlying (∞,1)-site is the étale $(\infty,1)$-site;
the admissible morphisms are the étale morphisms.
This is (Lurie, def. 4.3.13).
The geometry generated by the étale pregeometry $\mathcal{T}_{et}$ is the étale geometry $\mathcal{G}_{et}$.
This is (Lurie, prop. 4.3.15).
étale $(\infty,1)$-site
In its presentation as a model site the étale $(\infty,1)$-site is given in definition 2.2.2.12 of
A discussion in the context of structured (∞,1)-toposes is
See also