Anomalies on the Lattice, Homotopy of Quantum Cellular Automata, and a Spectrum of Invertible States

We develop a rigorous topological theory of anomalies on the lattice, which are obstructions to gauging global symmetries and the existence of trivial symmetric states. We also construct $\mathrm{\Omega }$-spectra of a class of invertible states and quantum cellular automata, which allows us to classify both anomalies and symmetry protected topological phases up to blend equivalence.