## $\mathbf{\Pi}_inf$-closed morphisms +-- {: .num_lemma} ###### Definition ($\mathbf{\Pi}_inf$-closed morphism) =-- +-- {: .num_lemma} ###### Definition ($\mathbf{\Pi}_inf$-closed object) =-- +-- {: .num_lemma} ###### Theorem (Formally étale subslices are coreflecive) =-- ## Models ### Étale groupoids +-- {: .num_lemma} ###### Theorem (Classical étale groupoids) =-- +-- {: .num_lemma} ###### Theorem (Formally étale $\infty$-groupoids are étale simplicial manifolds) =-- #### $\infty$-orbifolds +-- {: .num_lemma} ###### Definition ($\infty$-orbifold) =-- +-- {: .num_lemma} ###### Theorem (De Rham theorem for $\infty$-orbifolds) =-- +-- {: .num_lemma} ###### Observation (Inertia $\infty$-orbifold) =-- ### $U$-modelled higher manifolds +-- {: .num_lemma} ###### 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. =-- +-- {: .num_lemma} ###### Definition ($U$-modelled $\infty$-manifold) =--