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1 vote

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

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. …
Manuel Bärenz's user avatar
4 votes
0 answers
88 views

Are there any dominant pivotal functors such that the regular representation is not mapped o...

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms. A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes calle …
Manuel Bärenz's user avatar
9 votes
2 answers
399 views

What is a true invariant of $G$-crossed braided fusion categories?

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 …
Manuel Bärenz's user avatar
2 votes
1 answer
477 views

When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?

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 …
Manuel Bärenz's user avatar
2 votes
Accepted

When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?

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 …
Manuel Bärenz's user avatar
8 votes
0 answers
400 views

Which Drinfeld centers are balanced monoidal, i.e. have a twist?

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 …
Manuel Bärenz's user avatar
1 vote

Do all 3D TQFTs come from Reshetikhin-Turaev?

If your TQFT does not extend to the circle, (EDIT: and you're willing to generalise to lax TQFTs,) then the answer is no. A reference is: https://arxiv.org/abs/1408.0668 The proof goes by an explicit …
Manuel Bärenz's user avatar
12 votes
2 answers
714 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 o …
Manuel Bärenz's user avatar
17 votes
2 answers
1k views

How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?

Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original p …
Manuel Bärenz's user avatar
8 votes
1 answer
357 views

What's the (monoidal) image of a monoidal functor?

For an ordinary functor $F\colon \mathcal{C} \to \mathcal{D}$ of categories, there is a construction $\operatorname{im} F$, the image of $F$, which is again a category, and $F$ factors through that im …
Manuel Bärenz's user avatar
10 votes
2 answers
598 views

What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum ...

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the gro …
Manuel Bärenz's user avatar
4 votes
3 answers
956 views

When does a monoidal functor between ribbon categories preserve cups and caps, but not neces...

Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the repre …
Manuel Bärenz's user avatar
5 votes
0 answers
417 views

How does the relative Drinfeld center interact with the relative Deligne tensor product?

Let $\mathcal{C}$ be a fusion category, and $\mathcal{M}, \mathcal{N}$ semisimple $(\mathcal{C}, \mathcal{C})$-bimodule categories. The left $\mathcal{C}$-action is denoted as $- \triangleright - \col …
Manuel Bärenz's user avatar
13 votes
1 answer
638 views

A cohomology theory for fusion categories

It is well known that for a finite group $G$, the associator of the fusion category of $G$-graded $k$-vector spaces is given by an element of $H^3(G,k^*)$, up to equivalence of categories. ($k^*$ is t …
Manuel Bärenz's user avatar