  We develop a theoretical framework for the classification and construction of
symmetry protected topological (SPT) phases, which are a special class of
zero-temperature phases of strongly interacting gapped quantum many-body
systems that exhibit topological properties. The framework unifies various
proposals for the classification of SPT phases, including the group
(super-)cohomology proposal, the (spin-)cobordism proposal, the Freed-Hopkins
proposal, and the Kitaev proposal. The power of the framework is demonstrated
in a number of applications: (1) the classification and construction of 3D
fermionic SPT phases in Wigner-Dyson classes A and AII with glide symmetry, (2)
the classification and construction of 3D bosonic SPT phases with space-group
symmetries for all 230 space groups, (3) the derivation of a Mayer-Vietoris
sequence relating the classification of SPT phases with and without reflection
symmetry, and (4) an interpretation of the structure of general crystalline SPT
phases via the Atiyah-Hirzebruch spectral sequence. The framework is based on
Kitaev's idea that short-range entangled states form what are known as
$\Omega$-spectra in the sense of algebraic topology, and models the
classification of SPT phases by what are called generalized cohomology
theories.
