In the study of dualizability of tensor categories, Brochier-Jordan-Snyder observe that the tensor product of two (compact-)rigid monoidal categories may fail to be (compact-)rigid, and hence the collection of such categories fails to form a well-behaved higher category, in which fully extended topological field theories could take values. To remedy this, one weakens the definition of a tensor category and instead requires only rigidity of (compact) projective objects, known as cp-rigidity.

In this talk, I will present some initial structural results for module categories over finite cp-rigid categories, which we term pre-tensor categories, following Décoppet. This includes variants of the standard reconstruction results via algebra and bimodule objects, a description of exact module categories, and, perhaps most importantly, a characterization of pre-tensor categories which are Morita equivalent to (rigid, finite) tensor categories. Higher-categorical applications include a characterization of fully dualizable pre-tensor categories, and of Witt equivalence as equivalence internal to the Morita 4-category of braided tensor categories.

This talk is based on work in progress joint with Thibault Décoppet.

TBA

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