Spahn étale types (Rev #19)

Contents

\Box-closed morphisms

Definition

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

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

is a pullback square.

Theorem

The class of \Box-closed morphisms CC satisfies the following closure properties:

(1) Every equivalence is \Box-closed.

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

(3) The left cancellation property is satisfied: If h=gfh=g\circ f and hh and gg are \Box-closed, then so is ff.

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

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

Remark

A class of \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 reflective andcoreflecive)

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

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

(2) By the Giraud-Lurie axioms C/XC/X and (C/X) (C/X)_\Box are (,1)(\infty,1)-toposes.

(3) In particular C :=(C/*) CC_\Box:=(C/*)_\Box\hookrightarrow C is a reflective and coreflective subtopos.

Remark

(relation of reflective subcategories and reflective factorization systems)

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

Let 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

For the classs EE of Π inf\mathbf{\Pi}_inf-closed morphisms in CC 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.

(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 in 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) A manifold 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 and Definition

(1) In a cohesive (,1)(\infty,1)-topos HH, n\mathbb{R}^n can be defined solely in terms of the internal logic of HH.

(2) The previous theorem suggests to call an object XHX\in H an UU-modelled hausdorff \infty-manifold or an UU-modelled étale \infty-manifold if there is an étale atlas UXU\to X of XX.

\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)

As a corollary to the De Rham Theorem for étale objects we obtain the de Rham Theorem for \infty-orbifolds.

Observation and Definition (Inertia \infty-orbifold)

The free loop space object of an \infty-orbifold is an \infty-orbifold and is called the inertia \infty-orbifold of XX.

Models

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

(In a reflective subcategory CDC\hookrightarrow D a CC-morphism is a monomorphism iff it is a monomorphism in DD.

In a coreflective subcategory CDC\hookrightarrow D a CC-morphism is a epimorphism iff it is a epimorphism in DD.

The property of being a monomorphism is preserved by right adjoint functors.

The property of being a epimorphism is preserved by left adjoint functors.

)

Lemma

Let f:XYf:X\to Y be a morphism in HH.

Étale groupoids

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

Revision on December 11, 2012 at 00:04:37 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.