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We present an overview of the notions of exact sequences of Hopf algebras and tensor categories and their connections. We also present some examples illustrating their main features; these include simple fusion categories and a natural…
We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow the category of monoids for the other with…
We generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ``conformal vertex algebra'' or even more generally,…
This is the first of a pair of papers where we construct and investigate a closed monoidal structure on the category of generalized algebraic theories (in the sense of Cartmell). In the present text, as a starting point, we define the…
Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category…
Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored,…
This paper introduces the concept of distorted monoidal categories, a generalization of monoidal and braided monoidal categories that supports non-reversible and direction-sensitive tensor structures. Unlike the classical setting, where the…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact $\Bbbk$-linear monoidal functor can be lifted…
Following Nori's original idea we here provide certain motivic categories with a canonical tensor structure. These motivic categories are associated to a cohomological functor on a suitable base category and the tensor structure is induced…
We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining exact closed tensor categories of…
These are the lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of…
Let G be a unipotent algebraic group over an algebraically closed field k of characteristic p > 0 and let l be a prime different from p. Let e be a minimal idempotent in D_G(G), the braided monoidal category of G-equivariant (under…
We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains.
In this work we construct a compactly generated tensor-triangulated stable category for a large class of infinite groups, including those in Kropholler's hierarchy $\mathrm{LH}\mathfrak{F}$. This can be constructed as the homotopy category…
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…
A tensor extriangulated category is an extriangulated category with a symmetric monoidal structure that is compatible with the extriangulated structure. To this end we define a notion of a biextriangulated functor $\mathcal{A} \times…
In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors…
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs,…
This is the third part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part III), we introduce and study…