What are orbifolds with corners? What is the geometric definition of orbifolds with corners? Here “geometric" means that there is a definition in chapter 8 of the draft of Dominic Joyce's book D-manifolds and d-orbifolds: a theory of derived differential geometry (book website, direct pdf link), but that is too technical and abstract, and so I am trying to find a definition parallel to the definition of orbifolds in the normal way (locally it is an open set acted by a smooth group, etc.) or more intuitive than the definition in that book. If it is defined as an open set with boundary (its closure) acted by a group keeping the boundary invariant seems does not work, as the boundary point of a manifold with boundary is  itself a point in an orbifold (a manifold with boundary is an ordinary orbifold).
 A: Orbifolds with corners are defined by the same axioms as manifolds with corners and ordinary orbifolds: A topological $n$-dimensional orbifold with corners is a topological space $X$ (2nd countable and Hausdorff) equipped with a (maximal) "orbifold atlas" $\{U_i \ldots : i\in I\}$ consisting of open subsets $U_i\subset X$, open subsets $V_i\subset [0,\infty)^n$ and finite affine groups $\Gamma_i$ preserving $V_i$'s, together with homeomorphisms  $\phi_i: V_i/\Gamma_i\to U_i$, satisfying a long list of compatibility conditions which are identical to the ones for ordinary orbifolds with one important addition: Gluing maps $\psi_{ij}: V_i\to V_j$ preserve the boundary stratifications of $V_i$ and $V_j$ given by the product structure of $[0,\infty)^n$ (just as in the case of manifolds with corners).  
As a special case, a good orbifold with corners is the quotient $M/\Gamma$ of a topological manifold with corners $M$ by a properly discontinuous group action $\Gamma\times M\to M$, which is locally linearizable and is by automorphisms of the manifold with corners $M$.    
