Spahn étale types (Rev #2, changes)

Definition (Π inf\mathbf{\Pi}_inf-closed morphism) 1.1.

Definition (Π inf\mathbf{\Pi}_inf-closed object) 1.2.

Theorem (Formally étale subslices are coreflecive) 1.3.

Theorem (Classical étale groupoids) 2.1.

Theorem (Formally étale ∞\infty-groupoids are étale simplicial manifolds) 2.2.

Definition (∞\infty-orbifold) 2.3.

Theorem (De Rham theorem for ∞\infty-orbifolds) 2.4.

Observation (Inertia ∞\infty-orbifold) 2.5.

Theorem (Hausdorff manifold) 2.6. (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 (UU-modelled ∞\infty-manifold) 2.7.