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étale (infinity,1)-site

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Context

(∞,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Étale morphisms

étale morphism (submersion, immersion)

in topology

in differential geometry

in algebraic geometry

Higher geometry

higher geometry / derived geometry

Ingredients

Concepts

Constructions

Examples

Theorems

Contents

Idea

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.

Definition

Let k be a commutative ring. Let T be the Lawvere theory of commutative associative algebras over k.

Definition

Let

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

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

Proposition

Let sCAlg k=(TAlg) Δ 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 ∘ for the (∞,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).

Notation

For X∈∞CAlg k op we write 𝒪(X) for the corresponding object in ∞CAlg k and conversely for A∈∞CAlg k we write SpecA for the corresponding object in ∞CAlg k op.

So Spec𝒪X=X and 𝒪SpecA=A, by definition of notation.

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

π *:sCAlg k→CAlg k.\pi_* : sCAlg_k \to CAlg_k \,.
Definition

A morphism SpecA→SpecB in ∞CAlg k op is an étale morphism if

  1. The underlying morphism Specπ 0(A)→Specπ 0(B) is an étale morphism of schemes;

  2. for each i∈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.

Definition

The étale (∞,1)-site is the (∞,1)-site whose underlying (∞,1)-category is the opposite (∞,1)-category ∞CAlg k op and whose covering famlies {SpecA i→SpecB} i∈I are those collections of morphisms such that

  1. every SpecA i→SpecB is an étale morphism

  2. there is a finite subset J⊂I such that the underlying decategorified family {Specπ 0(A j)→Specπ 0(B)} j∈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).

Properties

Derived étale geometry

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

Definition

(étale pregeometry)

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

    which is the full subcategory of CAlg k on those objects A∈CAlg k for which there exists an étale morphism k[x 1,⋯,x n]→A from the polynomial algebra in n generators for some n∈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).

Definition

(étale geometry)

Let 𝒢 et be the geometry (for structured (∞,1)-toposes) given by

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

  • the admissible morphisms are the étale morphisms.

This is (Lurie, def. 4.3.13).

Theorem

The geometry generated by the étale pregeometry 𝒯 et is the étale geometry 𝒢 et.

This is (Lurie, prop. 4.3.15).

References

In its presentation as a model site the étale (∞,1)-site is given in definition 2.2.2.12 of

A discussion in the context of structured (∞,1)-toposes is in section 4.3 of

Revised on March 19, 2012 11:29:48 by Urs Schreiber (89.204.155.155)
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