We prove that a finite braided tensor category
is invertible in the
Morita
–category
of
braided tensor categories if and only if it is nondegenerate. This includes
the case of semisimple modular tensor categories, but also nonsemisimple
examples such as categories of representations of the small quantum group at
good roots of unity. Via the cobordism hypothesis, we obtain new invertible
–dimensional
framed topological field theories, which we regard as a nonsemisimple framed version
of the Crane–Yetter–Kauffman invariants, after the Freed–Teleman and Walker
constructions in the semisimple case. More generally, we characterize invertibility for
– and
–algebras in an arbitrary
symmetric monoidal
–category,
and we conjecture a similar characterization of invertible
–algebras for any
. Finally, we propose
the Picard group of
as a generalization of the Witt group of nondegenerate braided fusion categories, and
pose a number of open questions about it.
Keywords
braided tensor category, topological field theory, higher
categories