Spahn étale types (Rev #15, changes)

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$\mathbf{\Pi}_inf$ -closed morphisms
Derived structures and models
- $U$ -modelled higher manifolds
- $\infty$ -orbifolds
Models
- Étale groupoids

$\mathbf{\Pi}_inf$ -closed morphisms

Definition

Let $† □$ ~~\dagger~~ \Box be an idempotent monad on a presentable $(\infty,1)$ -category $C$ . A morphism $f:X\to Y$ is called $† □$ ~~\dagger~~ \Box-closed if

\begin{matrix} X & \to & † □ X \\ ↓^{f} & ↓^{† □ f} \\ Y & \to & † □ Y \end{matrix}

\array{ X&\to ~~&\dagger~~ &\Box X \\ \downarrow^f ~~&&\downarrow^{\dagger~~ &&\downarrow^{\Box f} \\ Y&\to& ~~\dagger~~ \Box Y }

is a pullback square.

Theorem

The class of $† □$ ~~\dagger~~ \Box-closed morphisms $C$ satisfies the following closure properties:

(1) Every equivalence is $† □$ ~~\dagger~~ \Box-closed.

(2) The composite of two $† □$ ~~\dagger~~ \Box-closed morphisms is $† □$ ~~\dagger~~ \Box-closed.

(3) The left cancellation property is satisfied: If $h=g\circ f$ and $h$ and $g$ are $† □$ ~~\dagger~~ \Box-closed, then so is $f$ .

(4) Any retract of a $† □$ ~~\dagger~~ \Box-closed morphism is $† □$ ~~\dagger~~ \Box-closed.

(5) The class is closed under pullbacks which are preserved by $† □$ ~~\dagger~~ \Box.

Remark

A class of $† □$ ~~\dagger~~ \Box-closed morphism which is closed under pullback is an admissible structure defining a geometry in the sense of Lurie’s DAG.

Theorem (Formally étale subslices are coreflecive)

Let $C/X$ be a slice of $C$ . The full sub- $(\infty,1)$ -category $(C / X)_{† □} \overset{ι}{↪} C / X$ ~~(C/X)_\dagger\stackrel{\iota}{\hookrightarrow}~~ (C/X)_\Box\stackrel{\iota}{\hookrightarrow} C/X on those morphisms into $X$ which are $† □$ ~~\dagger~~ \Box-closed is reflective and coreflective; i.e. $\iota$ fits into an adjoint triple

(C / X)_{† □} \overset{\overset{L}{\leftarrow}}{\overset{↪}{\underset{Et}{\leftarrow}}} (C / X) .

~~(C/X)_\dagger~~ (C/X)_\Box \stackrel{\overset{L}{\leftarrow}}{\stackrel{\hookrightarrow}{\underset{Et}{\leftarrow}}} (C/X) \,.

In particular $C_{† □} : = (C / *)_{† □} ↪ C$ ~~C_\dagger:=(C/*)_\dagger\hookrightarrow~~ C_\Box:=(C/*)_\Box\hookrightarrow C is reflective and coreflective.

Remark

(relation of reflective subcategories and reflective factorization systems)

Example ( $\mathbf{\Pi}_inf$ -closed morphism)

Let $H$ be a cohesive $(\infty,1)$ -topos equipped with infinitesimal cohesion

(i_!\dashv i^*\dashv i_*):H\stackrel{i_*}{\to} H_th

Then the class of formally étale morphisms in $H$ equals the class of $\mathbf{\Pi}_inf:=i_*i^*$ -closed morphisms in $H_th$ which happen to lie in $H$ .

Theorem

For the classs $E$ of $\mathbf{\Pi}_inf$ -closed morphisms in $C$ we have in addition to the above closure properties also the following ones:

(1) If in a pullback square in $C$ the left arrow is in $E$ and the bottom arrow is an effective epimorphism, then the right arrow is in $E$ .

(2) Every morphism $D\to *$ from a discrete object to the terminal object is in $E$ .

(3) $E$ is closed under colimit (taken in the arrow category).

(4) $E$ is closed under forming diagonals.

Definition ( $\mathbf{\Pi}_inf$ -closed object)

An object of $H$ is called formally étale object if there is a formally étale (effective) epimorphism (called atlas) from a $0$ -truncated object into $X$ .

Theorem (De Rham theorem for formally étale objects)

The de Rham theorem holds for any formally étale object for which the de Rham theorem holds level-wise in regard to the Cech nerve induced from the atlas.

Derived structures and models

$U$ -modelled higher manifolds

Definition (Cover, Atlas)

Let $U\to X$ be a morphism of an $(\infty,1)$ -category.

(1) We call $U\to X$ a cover of $X$ if it is an effective epimorphism.

(2) We call $U\to X$ a relative cover wrt. a class $M$ of morphisms if its pullback along every morphism in $M$ is a cover of $U$ and lies in $M$ .

(3) We call $U\to X$ a $n$ -atlas of $X$ , if it is a cover and $U$ is $n$ -truncated. A $0$ -atlas we call just an atlas.

Theorem (Hausdorff manifold)

(1) $X$ is a paracompact if there is a jointly epimorphic set of monomorphisms $\phi_i:\mathbb{R}^n\to X$ such that the corresponding Cech groupoid $\zeta_\phi$ is degree-wise a coproduct of copies of $\mathbb{R}^n$ .

(2) $X$ is hausdorff if $\zeta_\phi$ is moreover étale.

Remark

The previous theorem suggests that the theory of higher orbifolds is a natural continuation of that of hausdorff manifolds. Hence we could them just call higher hausdorff manifolds.

Definition ( $U$ -modelled $\infty$ -manifold)

$\infty$ -orbifolds

Definition ( $\kappa$ -compact object, $\kappa$ -compact cover, relatively $\kappa$ -compact atlas)

Let $C$ be an $(\infty,1)$ -category. Let $U\to X$ be a morphism.

(1) $U$ is called $\kappa$ -compact if $C(X,-)$ preserves $\kappa$ -filtered colimits.

(2) $U\to X$ is called $\kappa$ -compact cover if it is a cover and $U$ is $\kappa$ -compact.

(3) $U\to X$ is called relative $\kappa$ -compact cover if it is a relative cover wrt. all morphisms with $\kappa$ -compact domain.

Remark

The class of relative $\kappa$ -compact covers is closed under composition, pullbacks, and contains all isomorphisms.

Definition ( $\infty$ -orbifold)

An $\infty$ -orbifold is defined to be a groupoid object in $H$ posessing a relative $\kappa$ -compact atlas which is also $\mathbf{\Pi}_inf$ -closed.

Corollary (De Rham theorem for $\infty$ -orbifolds)

Observation (Inertia $\infty$ -orbifold)

Models

The $(\infty,1)$ -topos of synthetic differential $\infty$ -groupoids is an infinitesimal cohesive neighborhood of the $(\infty,1)$ -topos of smoooth $\infty$ -groupoids.

Étale groupoids

Lemma

Theorem (Classical étale groupoids)

Theorem (Formally étale $\infty$ -groupoids are étale simplicial manifolds)

Revision on December 10, 2012 at 00:02:45 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn étale types (Rev #15, changes)

Contents

Π inf\mathbf{\Pi}_inf-closed morphisms

Definition

Theorem

Remark

Theorem (Formally étale subslices are coreflecive)

Remark

Example (Π inf\mathbf{\Pi}_inf-closed morphism)

Theorem

Definition (Π inf\mathbf{\Pi}_inf-closed object)

Theorem (De Rham theorem for formally étale objects)

Derived structures and models

UU-modelled higher manifolds

Definition (Cover, Atlas)

Theorem (Hausdorff manifold)

Remark

Definition (UU-modelled ∞\infty-manifold)

∞\infty-orbifolds

Definition (κ\kappa-compact object, κ\kappa-compact cover, relatively κ\kappa-compact atlas)

Remark

Definition (∞\infty-orbifold)

Corollary (De Rham theorem for ∞\infty-orbifolds)

Observation (Inertia ∞\infty-orbifold)

Models

Étale groupoids

Lemma

Theorem (Classical étale groupoids)

Theorem (Formally étale ∞\infty-groupoids are étale simplicial manifolds)

$\mathbf{\Pi}_inf$ -closed morphisms

Example ( $\mathbf{\Pi}_inf$ -closed morphism)

Definition ( $\mathbf{\Pi}_inf$ -closed object)

$U$ -modelled higher manifolds

Definition ( $U$ -modelled $\infty$ -manifold)

$\infty$ -orbifolds

Definition ( $\kappa$ -compact object, $\kappa$ -compact cover, relatively $\kappa$ -compact atlas)

Definition ( $\infty$ -orbifold)

Corollary (De Rham theorem for $\infty$ -orbifolds)

Observation (Inertia $\infty$ -orbifold)

Theorem (Formally étale $\infty$ -groupoids are étale simplicial manifolds)