6 - Boundaries, corners, and the corner functor

Summary

If X is a manifold with corners of dimension n, the boundary dX is a manifold with corners of dimension n-1, and the k-fold boundary d^kX a manifold with corners of dimension n-k. The ‘k-corners’ C_k(X) is d^kX/S_k, also a manifold with corners of dimension n-k. The ‘corners’ C(X) is the disjoint union of all C_k(X), a manifold with corners of mixed dimension.

A smooth map f : X -> Y of manifolds with corners need not map dX -> dY, that is, boundaries are not functorial. But there is a natural map C(f) : C(X) -> C(Y), which need not map C_k(X) -> C_k(Y). This is the ‘corner functor’ for manifolds with corners.

We extend the corner functor to $C^\infty$-schemes with corners. It is right adjoint to the inclusion functor from interior $C^\infty$-schemes with corners to all $C^\infty$-schemes with corners, and so is canonically determined by the notion ‘interior’. Using the corner functor we define boundaries and corners of ‘firm’ $C^\infty$-schemes with corners.

We use the corner functor to study fibre products of $C^\infty$-schemes with corners, and show that b-transverse fibre products of manifolds with (g-)corners map to fibre products of $C^\infty$-schemes with corners.