Spahn étale types (Rev #3)

Definition

Let $\dagger$ be a 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 $\dagger$.

Lemma

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 ($\mathbf{\Pi}_inf$-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 $\mathbf{\Pi}_inf$-closed morphisms in $H_th$ which happen to lie in $H$.

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.

Theorem (Classical étale groupoids)

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

Definition ($\infty$-orbifold)

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

Observation (Inertia $\infty$-orbifold)

Theorem (Hausdorff manifold)

(1) $X$ is a paracompact if there is a 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.

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