Spahn étale types (Rev #5, changes)

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

Contents

Definition

Let †\dagger be a monad on a presentable (∞,1)(\infty,1)-category CC. A morphism f:X→Yf: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=g∘fh=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.

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

A Let class of†\dagger -closed morphism be which a is monad closed on under a pullback presentable is anadmissible structure(∞,1)(\infty,1) -category defining ageometryCC . in A the morphism sense of Lurie’s DAG.f:X→Yf: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 (Formally étale subslices are coreflecive)

Let The class ofC†/X C/X \dagger -closed be morphisms a slice ofCC . The satisfies full the sub- following closure properties:(∞,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) †←Et↪←L(C/X). (C/X)_\dagger \stackrel{\overset{L}{\leftarrow}}{\stackrel{\hookrightarrow}{\underset{Et}{\leftarrow}}} (C/X) \,.

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

In (2) particular The composite of twoC ††:=(C/*) †↪C C_\dagger:=(C/*)_\dagger\hookrightarrow \dagger C -closed morphisms is reflective and coreflective.†\dagger-closed.

(3) The left cancellation property is satisfied: If h=g∘fh=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.
Example Remark (Π inf\mathbf{\Pi}_inf-closed morphism)

Let A class ofH† H \dagger -closed be morphism a which cohesive is closed under pullback is an(∞,1)(\infty,1)admissible structure -topos equipped defining with a infinitesimal cohesiongeometry in the sense of Lurie’s DAG.

(i !⊣i *⊣i *):H→i *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 (Formally étale subslices are coreflecive)

For Let the classs E C/X E C/X be a slice ofΠ infC \mathbf{\Pi}_inf C -closed . morphisms The in full sub-C(∞,1) C (\infty,1) -category we have in addition to the above closure properties also the following ones:(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

(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.

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

(2) In Every particular morphismDC † → :=(C/*) †↪C D\to C_\dagger:=(C/*)_\dagger\hookrightarrow * C from a discrete object to the terminal object is in reflective and coreflective.EE.

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

(4) EE is closed under forming diagonals.
Definition Example (Π inf\mathbf{\Pi}_inf -closed object) morphism)

An Let object ofHH is be called a cohesiveformally étale object(∞,1)(\infty,1) -topos if equipped there with is infinitesimal a cohesion formally étale (effective) epimorphism (calledatlas) from a 00-truncated object into XX.

(i !⊣i *⊣i *):H→i *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

UU-modelled higher manifolds
Theorem (Hausdorff manifold)

(1) XX is a paracompact if there is a set of monomorphisms ϕ i:ℝ n→X\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 ℝ n\mathbb{R}^n.

(2) XX is hausdorff if ζ ϕ\zeta_\phi is moreover étale.
Definition (UU-modelled ∞\infty-manifold)

∞\infty-orbifolds
Definition (Compact object)
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)

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

UU-modelled higher manifolds
Theorem (Hausdorff manifold)

(1) XX is a paracompact if there is a set of monomorphisms ϕ i:ℝ n→X\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 ℝ n\mathbb{R}^n.

(2) XX is hausdorff if ζ ϕ\zeta_\phi is moreover étale.
Definition (UU-modelled ∞\infty-manifold)

Revision on December 3, 2012 at 21:17:19 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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