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

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 coreflecive)

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

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

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

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.

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 $X$.

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 $X$.

Lemma

Theorem (Classical étale groupoids)

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