6
votes
2answers
239 views

Is “being a modular category” a universal or categorical/algebraic property?

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 ...
2
votes
0answers
193 views

Braidings and Isomorphism Classes in a Monoidal Category

Let $X$ be an object in a monoidal category $({\cal C}, \otimes)$, and $\gamma:X \otimes X \to X \otimes X$ a braiding (that is to say a morphism in ${\cal C}$ from $X \otimes X$ to itself that ...
5
votes
2answers
402 views

Module categories over symmetric/braided monoidal categories

Given an algebraically closed field $k$ and a finitely generated commutative $k$-algebra $A$, all simple modules over $A$ are 1-dimensional What is the analogous statement for symmetric monoidal ...
5
votes
1answer
140 views

Extending braidings to tensor powers

Given a braiding $\Psi: X \otimes Y \to Y \otimes X$ for two objects $X,Y$ in a monoidal category, it seems reasonable to assume that $\Psi$ extends uniquely to a braiding $X^k \otimes Y^l \to Y^l ...
6
votes
2answers
394 views

Does the dual of an object with trivial symmetry also have trivial symmetry?

Let $C$ be a symmetric monoidal category. I am interested in objects $X \in C$ such that the symmetry $S_{X,X} : X \otimes X \cong X \otimes X$ is equal to the identity. There are many examples of ...
6
votes
3answers
646 views

180˚ vs 360˚ Twists in String Diagrams for Ribbon Categories

Ribbon categories are braided monoidal categories with a twist or balance, $\theta_B:B\to B$, which is a natural transformation from the identity functor to itself. In the string diagram calculus for ...
6
votes
4answers
2k views

What is the universal enveloping algebra?

Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). ...