: In these lectures, I will describe recent advances in defining the space of gapped lattice systems and the implications for the classification of gapped phases of matter. Until about 10 years ago, it has been widely believed that such phases can be classified using Topological Quantum Field Theory (TQFT). While recent developments showed that this is too optimistic, it is plausible that Short-Range Entangled (SRE) phases are related to invertible TQFT. By the cobordism conjecture, this suggests that SRE phases are classified by the homotopy groups of a certain Omega-spectrum. In turn, this implies that infinite-dimensional spaces of SRE systems carry cohomology classes which generalize the Berry curvature. I will explain how to construct these “higher Berry classes” and their equivariant versions, including the Hall conductance and its nonabelian analogs. The key ingredient in the construction is a differential graded Frechet-Lie algebra attached to any infinite-volume lattice system.