  A typical crystal is a finite piece of a material which may be invariant
under some point symmetry group. If it is a so-called intrinsic higher-order
topological insulator or superconductor then it displays boundary modes at
hinges or corners protected by the crystalline symmetry and the bulk topology.
We explain the mechanism behind that using operator K-theory. Specifically, we
derive a groupoid $C^\ast$-algebra that 1) encodes the dynamics of the
electrons in the infinite size limit of a crystal; 2) remembers the boundary
conditions at the crystal's boundaries, and 3) accepts a natural action by the
point symmetries of the atomic lattice. The filtrations of the groupoid's unit
space by closed subsets that are invariant under the groupoid and point group
actions supply equivariant co-filtrations of the groupoid $C^\ast$-algebra. We
show that specific derivations of the induced spectral sequences in twisted
equivariant K-theories enumerate all non-trivial higher-order bulk-boundary
correspondences.
