The study of module categories over general monoidal categories extends beyond the rigid, -linear, semisimple setting, though many results rely on additional assumptions. Here's a structured overview addressing your questions:

1. Module Categories and Bimodule Categories

A left module category over a monoidal category is a category with a functor $\otimes: ...

There is a fairly general theory beyond the rigid semisimple case, and (under suitable finiteness/exactness hypotheses) it looks much like the fusion-category story you already know.

1. The bicategory of module categories and the relative tensor product

• General formalism. For any monoidal category C, one can view C as a pseudomonoid in the monoidal bicategory Prof of categories and profunctors...

Below is a summary (with precise references) of what is presently known when one removes the words “semisimple” and even “fusion’’ from the usual theory of module-categories.

1. Module categories (= actegories) in complete generality

The notion of a left, right or bi-...

During the last fifteen years most of the technology that was first developed for fusion categories has been carried over – with essentially the same formal properties – to monoidal categories that are not assumed to be semi-simple. The right generality in which essentially all the “Brauer–Picard’’ statements work is that of a finite tensor category over a field k, i.e. a k–linear abelian monoid...

Here is a sketch of what is known in essentially arbitrary monoidal‐category–theoretic generality. Nothing really “mystical’’ happens once you drop semisimplicity – you simply rediscover the usual bicategorical/Morita‐theory machinery, now incarnated inside Cat (or inside whatever ambient 2-category you like).

1. Modules and bimodules as 1-cells in a bicategory of monoids
   – A monoidal...

This is an excellent question about the generalization of module category theory beyond the fusion category setting. Let me address the various aspects of your question.

General Theory of Module Categories

While much of the literature does focus on the fusion category case, there has been significant work on module categories in more general settings:

1. Basic Framework: The theory of mo...

Excellent question. You've hit on a very active and important area of research that extends the beautiful theory developed for fusion categories to more general settings. The short answer is yes, a lot of work has been done, and the picture becomes significantly richer and more complex once you drop the assumptions of semisimplicity and rigidity.

Let's break down your questions and survey the...

Module Categories Over General Monoidal Categories

The question explores what's known about module categories over general monoidal categories beyond the well-studied case of fusion categories (rigid k-linear semisimple).

Current State of Knowledge

While the definition of module categories makes sense in general settings, most deep results are in the fusion category context. Since Ostrik's...

Hmm, the core question seems to be about generalizing Brauer group concepts to monoidal categories without assuming ...