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Generalising Omar's answer, braided monoidal categories are (a special case of) tricategories with one 1-morphism. Symmetric monoidal categories are tetracategories (whatever that is) with one 2-morph …

I'm surprised nobody has mentioned Jeff Eggers "On involutive monoidal categories", which generalises Peter Selingers work, I guess. It's an excellent article, and it connects to other special cases. …

Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different. Premodular categories In braided spheri …

Definition. An invariant of a (spherical) fusion category with extra structure is a number or a set or tuple of numbers preserved under (appropriate) equivalences. (Spherical) fusion categories have …

I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is n …

tetrapharmakon's hint is excellent. The article he refers to (and the earlier article http://arxiv.org/abs/math/0604180 by some of the same authors) define "quasi-triangular Hopf monads" that are an a …

A twist is an automorphism $\theta$ of the identity functor of a monoidal category with braiding $c$, such that $\theta_{X \otimes Y} = c_{Y,X} c_{X,Y} (\theta_X \otimes \theta_Y)$. A braided monoidal …

Edit: I used to believe that there is a possible generalisation stemming from work of Altschüler and Bruguières. (See Appendix C in Drinfeld, Gelaki, Nikshych, Ostrik - On braided fusion categories I) …

In Müger's article "Conformal Field Theory and Doplicher-Roberts Reconstruction", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore eve …

A semisimple braided category with duals is called modular when a certain matrix $S$ is invertible. The components $S_{AB}$ are indexed by (isomorphism classes of) simple objects of the category and o …

Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural transformatio …

As far as I know, low-dimensional semisimple Hopf algebras are classified (along with non-semisimple ones) up to dimension 60, with the first example of a semisimple Hopf algebra not coming from a fin …

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in …

Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations: $$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$ $$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12 …

In a sense, this is a follow up on this question, but one PhD programme later. Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of obje …