Spahn étale types (Rev #2)

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

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

Theorem (Formally étale subslices are coreflecive)

Theorem (Classical étale groupoids)

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

Definition ($\infty$-orbifold)

Theorem (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)