nLab étale (infinity,1)-site



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

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos

Étale morphisms

Higher 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 opsCAlg_k^{op} of commutative simplicial algebras over a commutative ring kk, whose covering families are essentially those which under decategorification become coverings in the étale site of ordinary kk-algebras.


Let kk be a commutative ring. Let TT be the Lawvere theory of commutative associative algebras over kk.



CAlg k:=TAlg \infty CAlg_k := T Alg_\infty

be the (∞,1)-category of ∞-algebras over TT regarded as an (∞,1)-algebraic theory.


Let sCAlg k=(TAlg) Δ opsCAlg_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 sCAlg_k^\circ for the (,1)(\infty,1)-category presentable (∞,1)-category. Then we have an equivalence of (∞,1)-categories

CAlg k(sCAlg k) . \infty CAlg_k \simeq (sCAlg_k)^\circ \,.

This is a special case of the general statement discussed at (∞,1)-algebraic theory. See also (Lurie, remark 4.1.2).


For XCAlg k opX \in \infty CAlg_k^{op} we write 𝒪(X)\mathcal{O}(X) for the corresponding object in CAlg k\infty CAlg_k and conversely for ACAlg kA \in \infty CAlg_k we write SpecASpec A for the corresponding object in CAlg k op\infty CAlg_k^{op}.

So Spec𝒪X=XSpec \mathcal{O} X = X and 𝒪SpecA=A\mathcal{O} Spec A = A, by definition of notation.

Notice from the discussion at model structure on simplicial algebras the homotopy group functor

π *:sCAlg kCAlg k. \pi_* : sCAlg_k \to CAlg_k \,.

A morphism SpecASpecBSpec A \to Spec B in CAlg k op\infty CAlg_k^{op} is an étale morphism if

  1. The underlying morphism Specπ 0(A)Specπ 0(B)Spec \pi_0(A) \to Spec \pi_0(B) is an étale morphism of schemes;

  2. for each ii \in \mathbb{N} the canonical morphism

    π i(A) π 0(A)π 0(B)π i(B) \pi_i(A) \otimes_{\pi_0(A)} \pi_0(B) \to \pi_i(B)

    is an isomorphism.


The étale (,1)(\infty,1)-site is the (∞,1)-site whose underlying (,1)(\infty,1)-category is the opposite (∞,1)-category CAlg k op\infty CAlg_k^{op} and whose covering famlies {SpecA iSpecB} iI\{Spec A_i \to Spec B\}_{i \in I} are those collections of morphisms such that

  1. every SpecA iSpecBSpec A_i \to Spec B is an étale morphism

  2. there is a finite subset JIJ \subset I such that the underlying decategorified family {Specπ 0(A j)Specπ 0(B)} jJ\{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. and as (Lurie, def. 4.3.3; def. 4.3.13).


Derived étale geometry

The following definition and theorem show how the étale (,1)(\infty,1)-site arises naturally from the étale 1-site, and naturally encodes the derived geometry induced by the étale site.


(étale pregeometry)

Let 𝒯 et\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

    (CAlg k sm) opCAlg k op, (CAlg_k^{sm})^{op} \hookrightarrow CAlg_k ^{op} \,,

    which is the full subcategory of CAlg kCAlg_k on those objects ACAlg kA \in CAlg_k for which there exists an étale morphism k[x 1,,x n]Ak[x^1, \cdots, x^n] \to A from the polynomial algebra in nn generators for some nn \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 𝒢 et\mathcal{G}_{et} be the geometry (for structured (∞,1)-toposes) given by

  • the underlying (∞,1)-site is the étale (,1)(\infty,1)-site;

  • the admissible morphisms are the étale morphisms.

This is (Lurie, def. 4.3.13).


The geometry generated by the étale pregeometry 𝒯 et\mathcal{T}_{et} is the étale geometry 𝒢 et\mathcal{G}_{et}.

This is (Lurie, prop. 4.3.15).


In its presentation as a model site the étale (,1)(\infty,1)-site is given in definition 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.