Spahn étale types

$\Box$-closed morphisms

Definition

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

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

is a pullback square.

Theorem

The class of $\Box$-closed morphisms $C$ 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 $h=g\circ f$ and $h$ and $g$ are $\Box$-closed, then so is $f$.

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

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

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

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

Remark

(relation of reflective subcategories and reflective factorization systems)

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 $\mathbf{\Pi}_inf:=i_*i^*$-closed morphisms in $H_th$ which happen to lie in $H$.

Theorem

For the classs $E$ of $\mathbf{\Pi}_inf$-closed morphisms in $C$ we have in addition to the above closure properties also the following ones:

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

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

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

(4) $E$ is closed under forming diagonals.

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.

Derived structures and models

$U$-modelled higher manifolds

Definition (Cover, Atlas)

Let $U\to X$ be a morphism in an $(\infty,1)$-category.

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

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

(3) We call $U\to X$ a $n$-atlas of $X$, if it is a cover and $U$ is $n$-truncated. A $0$-atlas we call just an atlas.

Theorem (Hausdorff manifold)

(1) A manifold $X$ is a paracompact if there is a jointly epimorphic 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.

Remark and Definition

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

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

$\infty$-orbifolds

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

Let $C$ be an $(\infty,1)$-category. Let $U\to X$ be a morphism.

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

(2) $U\to X$ is called $\kappa$-compact cover if it is a cover and $U$ is $\kappa$-compact.

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

Remark

The class of relative $\kappa$-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 $H$ posessing a relative $\kappa$-compact atlas which is also $\mathbf{\Pi}_inf$-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 $X$.

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 $(\infty,1)$-topos $H_th$ of synthetic differential $\infty$-groupoids is an infinitesimal cohesive neighborhood of the $(\infty,1)$-topos $H$-of smoooth $\infty$-groupoids. In this context we call $\Box$-closed morphisms also étale morphisms.

Definition

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

$$\diamond X\simeq *$$

Lemma (Tangent space)

For every $X\in H_th$ the$\Box$-unit $\eta^\Box_X$ evaluates in every infinitesimal smooth locus $D$ to the tangent bundle.

Proof

$$(\eta_X^\Box:X(D)\to (\Box X)(D))=([D,X]\to [D,\Box X]\simeq( [\diamond D,X]\simeq [*,X]\simeq X)$$

Proposition (submersion -, immersion -, and étale map of smooth paracompact manifolds)

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

$$X\to Y\times_{\Box Y} \Box X$$

is an epimorphism / monomorphism / isomorphism.

Proof

Every $Z\in H_th$ can be written as $Z=U\times D$ where $U\in H$ and $D$ is an infinitesimal smooth locus. We compute:

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

and analogously for $Y$.

Étale groupoids

Theorem

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

Then $f$ is $\Box$-closed.

Proof

$f$ is $\Box$-closed precisely if $U$ is equivalent to the homotopy pullback of

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

We compute this pullback equivalently by a ($1$-categorical) pullback by giving a fibration replacement of $\Box f$. Décalage $d_last:Dec (X)\to X$ is a fibration replacement of $f$ and since $\Box$ is a right Quillen functor $\Box d_last$ is a fibration replacement of $f$.

Since ($1$-categorial) pullbacks of simplicial objects are level-wise pullbacks it remains to show that for all $n$

$$U\to X_n\times_{(\Box X)_n}(\Box Dec(X))_n$$

is an isomorphism. By the previous Proposition, this is the case precisely if $d_lstn:Dec(X)_n\to X_n$ 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.

Proof

We compute the same pullback as in the proof of the previous Theorem but with the fibration replacement of $\Box f$ obtained by the factorization lemma. The fibration replacement of $f$ is the left vertical composite in

(diagram)

where $X\stackrel{e}{\to}X^I\stackrel{d_0,d_1}{\to}X\times X$ is the factorization of the codiagonal of $X$ through the pathspace object $X^I=X^{\Delta[1]}$ of $X$. Spelled out the pullback (diagram) which is by assumption a $1$-truncated groupoid interprets as

(list)

The factorization lemma says that

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

is a homotopy pullback iff

$$\array{ X^I\times_X U&\to&\Box X^I\times_X U \\ \downarrow&&\downarrow^{\Box \tilde{f}} \\ X&\stackrel{\eta^\Box_X}{\to}&\Box X }$$

is a $1$-categorial pullback. This means precisely that $f$ is $\Box$-closed iff $\tilde{f}$. Since the objects here are simplicial, the latter pullback is computed level-wise. By the previous Proposition the $1$-truncation of $f$ is $\Box$-closed iff the $1$ truncation of $\tilde{f}$ is a classical Lie groupoid.

$X$ being a classical Lie Groupoid means that

(description)

Remark (Étale simplicial manifold)

If $Mfd_et$ denotes a convenient category of manifolds with only étale maps as morphisms, and $G_\Delta^{op}$ is a simplicial object in this category, then Yoneda embedding of this object followed by sheafification yields an étale simplicial manifold.

Terminological remark on étaleness in algebraic geometry

One can refine the notion of $\Box$-closedness in the following way.

Definition

For some integer $k\ge -2$, a morphism $f:X\to Y$ in $H_th$ is called $\Box$-$k$-subclosed / $\Box$-closed / $\Box$-$k$-supclosed if the characterizing morphism

$$X\to Y\times_{\Box Y} \Box X$$

is $k$-truncted / isomorphism / $k$-connected.

Definition

An object $X\in H_th$ is called algebraically formally $k$-smooth / algebraically formally étale if the morphism $X\to *$ is $\Box$-$k$-supclosed / $\Box$-closed.

Remark

(1) Every $n$-truncated object is algebraically formally $k$-smooth.

(2) Every $n$-truncated object is algebraically formally étale if it is contractible.