Spahn étale types (Rev #22)

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

()(\diamond\dashv \Box)

What we described so far dualizes to the case of an idempotent comonad. Also the situation where we have adjoint modalities - i.e an adjoint pair ()(\diamond\dashv \Box) where \Box is an idempotent monad and \diamond is an idempotent comonad is of interest.

Synthetic differential geometry

We will describe the foundations of the theory of synthetic differential geometry in terms of adjoint modalities.

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 this context we call \Box-closed morphisms also étale morphisms.

Definition

An object XH thX\in H_th we call to be an infinitesimal smooth locus if X\diamond X is contractible.

X*\diamond X\simeq *
Lemma (Tangent space)

For every XH thX\in H_th the\Box-unit η X \eta^\Box_X evaluates in every infinitesimal smooth locus DD to the tangent bundle.

Proof
(η X :X(D)(X)(D))=([D,X][D,X]([D,X][*,X]X)(\eta_X^\Box:X(D)\to (\Box X)(D))=([D,X]\to [D,\Box X]\simeq( [\diamond D,X]\simeq [*,X]\simeq X)
Proposition (submersion -, immersion -, and étale map of smooth paracompact manifolds)

A morphism f:XYf:X\to Y of smooth paracompact manifolds in H thH_th is a submersion / imersion / étale morphism iff the induced morphism

XY× YXX\to Y\times_{\Box Y} \Box X

is an epimorphism / monomorphism / isomorphism.

Proof

Every ZH thZ\in H_th can be written as Z=U×DZ=U\times D where UHU\in H and DD is an infinitesimal smooth locus. We compute:

X(U×D)=[U×D,X]=[U,[D,X]] X(U×D)=[U×D,X]=[U,[D,X]]=[U,[D,X]]=[U,X]\array{ X(U\times D)=[U\times D, X]=[U,[D,X]] \\ \Box X(U\times D)=[U\times D,\Box X]=[U,[D,\Box X]]=[U,[\diamond D,X]]=[U,X]}

and analogously for YY.

Étale groupoids

Theorem

Let XX be an étale simplicial manifold being a Kan fibrant object in the projective model structure on simplicial presheaves equipped with an atlas f:UXf:U\to X.

Then ff is \Box-closed.

Proof

ff is \Box-closed precisely if UU is equivalent to the homotopy pullback of

U f X η X X\array{ &&\Box U \\ &&\downarrow^{\Box f} \\ X&\stackrel{\eta^\Box_X}{\to}&\Box X }

We compute this pullback equivalently by a (11-categorical) pullback by giving a fibration replacement of f\Box f. Décalage d last:Dec(X)Xd_last:Dec (X)\to X is a fibration replacement of ff and since \Box is a right Quillen functor d last\Box d_last is a fibration replacement of ff.

Since (11-categorial) pullbacks of simplicial objects are level-wise pullbacks it remains to show that for all nn

UX n× (X) n(Dec(X)) nU\to X_n\times_{(\Box X)_n}(\Box Dec(X))_n

is an isomorphism. By the previous Proposition, this is the case precisely if d lstn:Dec(X) nX nd_lstn:Dec(X)_n\to X_n is an étale map of manifolds. But this map is étale by the definition of “étale manifold” which we assumed.

Corollary (Classical étale groupoids)

An étale Lie groupoid possesses an étale atlas.

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

Revision on December 11, 2012 at 20:05:19 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.