Definition

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

is a pullback square.

Theorem

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

(4) Any retract of a $\dagger$-closed morphism is $\dagger$-closed.

(5) The class is closed under pullbacks which are preserved by $\dagger$.

Remark

A class of $\dagger$-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 $(C/X)_\dagger\stackrel{\iota}{\hookrightarrow} C/X$ on those morphisms into $X$ which are $\dagger$-closed is reflective and coreflective; i.e. $\iota$ fits into an adjoint triple

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

Example Remark ($\mathbf{\Pi}_inf$-closed morphism)

Let (relation of reflective subcategories and reflective factorization systems)$H$ be a cohesive $(\infty,1)$-topos equipped with infinitesimal cohesion

Theorem Example ($\mathbf{\Pi}_inf$-closed morphism)

For Let the classs$\mathrm{<span><del\; class="diffmod">\; E</del><ins\; class="diffmod">\; H</ins></span>}$ E H of be a cohesive${\Pi }_{\inf }(\mathrm{\infty },1)$ \mathbf{\Pi}_inf (\infty,1) -closed -topos morphisms equipped in with infinitesimal cohesion$C$ we have in addition to the above closure properties also the following ones:

Definition Theorem ($\mathbf{\Pi}_inf$-closed object)

An For object the of classs$\mathrm{<span><del\; class="diffmod">\; H</del><ins\; class="diffmod">\; E</ins></span>}$ H E is of calledformally étale object$\mathbf{\Pi}_inf$ -closed if morphisms there in is a formally étale (effective) epimorphism (calledatlas$C$ ) from we a have in addition to the above closure properties also the following ones:$0$-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 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) $X$ 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

The previous theorem suggests that the theory of higher orbifolds is a natural continuation of that of hausdorff manifolds. Hence we could them just call higher hausdorff manifolds.

Definition ($U$-modelled $\infty$-manifold)

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)

Observation (Inertia $\infty$-orbifold)

Lemma

Theorem (Classical étale groupoids)

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