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 of 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

The (1) previous In theorem suggests that the theory of higher orbifolds is a natural cohesive continuation of that of hausdorff manifolds. Hence we could them just call higher hausdorff manifolds.$(\infty,1)$-topos $H$, $\mathbb{R}^n$ can be defined solely in terms of the internal logic of $H$.

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

Let The class of relative$\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; \kappa </ins></span>}$ C \kappa -compact be covers an is closed under composition, pullbacks, and contains all isomorphisms.$(\infty,1)$-category. Let $U\to X$ be a morphism.

Remark Definition ($\infty$-orbifold)

The An class of relative$\kappa$$\infty$-orbifold -compact covers is closed defined under to composition, be pullbacks, a and groupoid contains object all in isomorphisms.$H$ posessing a relative $\kappa$-compact atlas which is also $\mathbf{\Pi}_inf$-closed.

Definition Corollary ( (De Rham theorem for$\infty$ -orbifold) -orbifolds)

An As a corollary to the$\infty$-orbifoldDe Rham Theorem for étale objects is we defined obtain to the be de a Rham groupoid Theorem object for in$H\mathrm{\infty }$ H \infty -orbifolds. posessing a relative$\kappa$-compact atlas which is also $\mathbf{\Pi}_inf$-closed.

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

Lemma

Theorem (Classical étale groupoids)

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