Spahn étale types (Rev #3, changes)

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

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

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

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

is a pullback square.

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

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

(1) Every equivalence is $\dagger$ -closed.

(2) The composite of two $\dagger$ -closed morphisms is $\dagger$ -closed.

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

(4) Any retract of a $\dagger$ -closed morphism is $\dagger$ -closed.

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

Theorem Lemma (Formally étale subslices are coreflecive)

Theorem (Formally étale subslices are coreflecive)

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

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

In particular $C_\dagger:=(C/*)_\dagger\hookrightarrow C$ is reflective and coreflective.

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 $\mathbf{\Pi}_inf$ -closed morphisms in $H_th$ which happen to lie in $H$ .

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.

Models

Étale groupoids

Theorem (Classical étale groupoids)

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

$\infty$ -orbifolds

Definition ( $\infty$ -orbifold)

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

Observation (Inertia $\infty$ -orbifold)

$U$ -modelled higher manifolds

Theorem (Hausdorff manifold)

(1) $X$ is a paracompact if there is a 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.

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

Revision on December 3, 2012 at 16:57:04 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn étale types (Rev #3, changes)

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

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

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

Theorem Lemma (Formally étale subslices are coreflecive)

Theorem (Formally étale subslices are coreflecive)

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

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