Rigid Convolution Structures

Abstract

A monoidal category is called a convolution monoidal category if it arises from linearizing a 2-Segal space. The goal of this thesis is to study for which 2-Segal spaces the induced convolution monoidal category is a multi-fusion category. With this aim, we show that multi-fusion categories admit an intrinsic description as rigid algebras in the symmetric monoidal 2-category of C-linear additive categories. We use this observation to define, by analogy, a derived version of a multi-fusion category as a rigid algebra in the symmetric monoidal (infinity,2)-category of stable infinity-categories. We show that examples of these arise as derived categories of multi-fusion categories and as categories of modules over smooth and proper E2-algebras. Afterward, we show that rigid algebras in the (infinity, 2)-category of spans are precisely given by those 2-Segal objects that are Čech-nerves. Together with our previous result, we use this to provide an answer to our initial question. To prove this result, we provide a description of bimodules in the infinity-category of spans as birelative 2-Segal objects. Furthermore, we introduce a notion of morphism between birelative 2-Segal objects that extends this classification to an equivalence of infinity-categories. We use this classification to construct examples of convolution monoidal structures that form derived multi-fusion categories and discuss some aspects of the associated fully extended TFTs. We finish by studying Grothendieck–Verdier-structures on convolution monoidal infinity-categories and by comparing them with rigid dualities.