# Questions tagged [braided-tensor-categories]

The tag has no usage guidance.

116 questions
Filter by
Sorted by
Tagged with
194 views

1 vote
101 views

### Braided R-matrices for finite action groupoids

1. Algebra from action groupoids Let $G$ be a finite group acting on a finite set $X$ from the right (denoted in element as $x^{g}$). We have an algebra (of the action groupoid) over $\mathbb{C}$: the ...
213 views

### What is the proof of the compatibility of a braiding with the unitors?

I am specifically referencing the property that, given a braided monoidal category with a braiding $c$ and left and right unitors $\lambda, \rho$, $$\lambda_A \circ c_{A,I}=\rho_{A},$$ for any ...
212 views

### Why is 'every braided monoidal category spacial'? [duplicate]

In his 2009 survey, Selinger ("A survey of graphical languages for monoidal categories") defines the notion of a 'spacial monoidal category', which (in his graphical calculus) allows one to ...
100 views

### An introductory reference for tensor networks

I found a good reference on Tensor Networks: https://arxiv.org/abs/1912.10049. But I need an introductory reference with detailed proofs on Tensor Networks. Do you know another reference?
96 views

### Connection between braided tensor categories and local systems on moduli of stable marked genus zero curves

I'm looking for references regarding an unpublished Deligne's manuscript "Une descrption de catégorie tressée (inspiré par Drinfeld)" and the subject it touches, that is described in the ...
393 views

### Braided monoidal category, example

Let $H$ be a cocommutative hopf algebra. Let $M$ be the category of $H$-bimodules. Does the category $M$ form a braided monoidal category with tensor product $\otimes_{H}$ ?
286 views

### Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$

$\newcommand\Mod{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between $$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$ The proof in "Tensor Categories ...
213 views

### Constructing the inverse of a braiding in a braided pivotal category

Assume we have a braided pivotal monoidal category. This means we assume the braiding $c$ to be a natural isomorphism. But looking at the corresponding string diagram, it seems to me as if we could ...
155 views

### Drinfeld center of $\mathrm{Mod}_R$

Let $R$ be a commutative ring and let $\mathrm{Mod}_R$ be the category of (left) $R$-modules. Question: Is it true that the categories $\mathcal{Z}(\mathrm{Mod}_R)$ and $\mathrm{Mod}_R$ are ...
1 vote
91 views

### Recovering the center of a monoid from the Drinfeld center

The Drinfeld center construction is intended to be a categorification of the center of a monoid. It seems to be folklore (eg this answer or this one) that when the Drinfeld center is taken over a ...
263 views

### Ordered logic is the internal language of which class of categories?

Wikipedia says: The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. "A Fibrational Framework for Substructural and Modal ...
102 views

### Can a braided fusion category have an order-2 Morita equivalence class which cannot be simultaneously connected and isomorphic to its opposite?

Let $\mathcal{B}$ be a braided fusion category over $\mathbb{C}$. Let me write $\mathrm{Alg}(\mathcal{B})$ for the set of isomorphism classes of unital associative algebra objects in $\mathcal{B}$, ...
146 views

### Categorical Morita equivalence implies equivalence of module categories?

Classically, two rings $R$ and $S$ are Morita equivalent if and only if any of the following is true ($R$-Mod) $\simeq$ ($S$-Mod). $S \simeq Hom_R(M,M)$, where $M$ is a finitely generated projective ...
253 views

### Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?

Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...
134 views

66 views

64 views

### Categorical construction of comodule category of FRT algebra

Let $\mathcal{B}$ denote the braid groupoid, with objects being non-negative integers $n \in \mathbb{Z}_{\geq 0}$ and morphisms $\mathcal{B}(n,n)=B_{n}$ given by the braid group. Let $\mathcal{C}$ be ...
1k views

### The tensor product of two monoidal categories

Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way? The motivation I am thinking of is two categories that are representation ...
171 views

285 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 ...
2k views

### What is a tensor category?

A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term tensor category is also used, and seems to mean a monoidal category with more ...
1 vote
147 views

### Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$

Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked. I know that there ...
648 views

### Enrichments vs Internal homs

Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor $$X \otimes -: \cal{C} \to \cal{C},$$ for ...
For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{*})^*$, for $*$ denoting dual? If so, what is a good example.