Speaker: Theo Johnson-Freyd (Perimeter Institute)
Title: Condensation and Idempotent Completion
Idempotent (aka Karoubi, aka Cauchy) completion appears throughout mathematics: for instance, it converts the category of free modules to the category of projective modules. I will explain the higher-categorical generalization of idempotent completion. I call it "condensation", because, as I will explain, if you start with a category of gapped phases of matter, then its idempotent completion consists of those phases that can be condensed from the phases you already have. In particular, if you start just with the vacuum phase, and idempotent complete, you recover a very large class of gapped phases, including the Turaev--Viro--Barrett--Westbury models. Moreover, every condensable-from-the-vacuum phase of matter is fully dualizable (i.e. determines a fully-extended TQFT), and conversely every condensable-from-the-vacuum TQFT has a commuting projector Hamiltonian model, and so one finds an equivalence between large classes of TQFTs and condensed phases. Based on joint work with Davide Gaiotto.
Chase, Room 319