Condensations in higher categories

TL;DR

This paper generalizes the Karoubi envelope to higher categories using 'condensations', linking topological phases of matter with extended topological field theories through a new categorical framework.

Contribution

It introduces a higher-categorical 'condensation' concept replacing idempotents, establishing a connection between topological phases and fully-dualizable objects.

Findings

Higher Karoubi envelope is the closure for absolute limits.

Condensations encode topological phase transitions.

Equivalence between gapped topological phases and topological field theories.

Abstract

We present a higher-categorical generalization of the "Karoubi envelope" construction from ordinary category theory, and prove that, like the ordinary Karoubi envelope, our higher Karoubi envelope is the closure for absolute limits. Our construction replaces the idempotents in the ordinary version with a notion that we call "condensations." The name is justified by the direct physical interpretation of the notion of condensation: it encodes a general class of constructions which produce a new topological phase of matter by turning on a commuting projector Hamiltonian on a lattice of defects within a different topological phase, which may be the trivial phase. We also identify our higher Karoubi envelopes with categories of fully-dualizable objects. Together with the Cobordism Hypothesis, we argue that this realizes an equivalence between a very broad class of gapped topological phases of matter and fully extended topological field theories, in any number of dimensions.