  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.
