this symmetry creates a natural ${\mathbb{Z}}_2$-graded algebraic structure. Since any operator, such as the Hamiltonian, can be thought of as a transformation acting on the Hilbert space, the study of all compatible Hamiltonians is thus equivalent to the study of all possible actions of the extended ${\mathbb{Z}}_2$-graded algebra on the Hilbert space; see Fig. \[fig:diag2\].

For each of the possible 32 crystallographic point group symmetries, we find the appropriate ${\mathbb{Z}}_2$-graded algebra and its corresponding “classifying space" of possible actions. The topological indices characterizing possible topological insulators and superconductors in any of the 10 AZ symmetry classes are the topological invariants of this classifying space, and these invariants are herein presented.

The majority of phases found by our classification paradigm correspond to novel crystalline topological insulators and superconductors. Nevertheless, a tremendous amount of work on various crystalline topological phases has recently been carried out by numerous research groups around the globe, we thus also compare our results to many phases that had already been analysed (see Sec. \[sec:discussion\]).

The rest of the paper is divided as follows: In Sec. \[sec:results\] we summarize our results which are presented in the tables throughout the paper. In Sec. \[sec:meth\] we present the physical paradigm 