Spahn étale types (Rev #14, changes)

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

Definition

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

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

is a pullback square.

Theorem

The class of \dagger-closed morphisms CC 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=gfh=g\circ f and hh and gg are \dagger-closed, then so is ff.

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

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

Remark

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

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

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

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

Let (relation of reflective subcategories and reflective factorization systems)HH be a cohesive (,1)(\infty,1)-topos equipped with infinitesimal cohesion

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

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

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

For Let the classs E H E H of be a cohesiveΠ inf(,1) \mathbf{\Pi}_inf (\infty,1) -closed -topos morphisms equipped in with infinitesimal cohesionCC we have in addition to the above closure properties also the following ones:

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

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

(2) Then Every the morphism class of formally étale morphisms in D H* D\to H * from equals a discrete object to the terminal class object of is inEΠ inf:=i *i * E \mathbf{\Pi}_inf:=i_*i^*-closed morphisms in H thH_th which happen to lie in HH.

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

(4) EE is closed under forming diagonals.

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

An For object the of classs H E H E is of calledformally étale objectΠ inf\mathbf{\Pi}_inf -closed if morphisms there in is a formally étale (effective) epimorphism (calledatlasCC ) from we a have in addition to the above closure properties also the following ones:00-truncated object into XX.

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

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

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

(4) EE is closed under forming diagonals.

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

An object of HH is called formally étale object if there is a formally étale (effective) epimorphism (called atlas) from a 00-truncated object into XX.

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

UU-modelled higher manifolds

Definition (Cover, Atlas)

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

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

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

(3) We call UXU\to X a nn-atlas of XX, if it is a cover and UU is nn-truncated. A 00-atlas we call just an atlas.

Theorem (Hausdorff manifold)

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

(2) XX is hausdorff if ζ ϕ\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 (UU-modelled \infty-manifold)

\infty-orbifolds

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

Let CC be an (,1)(\infty,1)-category. Let UXU\to X be a morphism.

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

(2) UXU\to X is called κ\kappa-compact cover if it is a cover and UU is κ\kappa-compact.

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

Remark

The class of relative κ\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 HH posessing a relative κ\kappa-compact atlas which is also Π inf\mathbf{\Pi}_inf-closed.

Corollary (De Rham theorem for \infty-orbifolds)
Observation (Inertia \infty-orbifold)

Models

The (,1)(\infty,1)-topos of synthetic differential \infty-groupoids is an infinitesimal cohesive neighborhood of the (,1)(\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 6, 2012 at 01:41:12 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.