# Questions tagged [braided-tensor-categories]

The braided-tensor-categories tag has no usage guidance.

124
questions

5
votes

1
answer

224
views

### Tannakian reconstruction for braided categories

Let $\mathcal{C}$ be a symmetric monoidal category. One can imagine a theorem
Tannakian reconstruction: If $\mathcal{B}$ is a braided monoidal category and $F:\mathcal{B}\to \mathcal{C}$ is a functor ...

0
votes

0
answers

61
views

### Is a Lagrangian subgroup of a metric group isomorphic to its quotient?

A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...

7
votes

1
answer

264
views

### Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?

What is the smallest group $G$ such that $\operatorname{Rep}(G)$ has a non-symmetric braiding (or just an easy example)?
I seem to remember a result classifying all universal $R$-matrices of $\mathbb ...

2
votes

0
answers

60
views

### Naturality of ribbon category twists

Tortile Tensor Categories by Shum defines a twist to be a natural transformation $\theta : \operatorname{Id} \to \operatorname{Id}$ satisfying some axioms. However, wikipedia and nLab worded the ...

8
votes

1
answer

280
views

### Is there a notion of "knot category"?

Consider a rigid braided monoidal category, with braiding $\beta_{x,y} : x \otimes y \cong y \otimes x$, and every object has a dual such that $\epsilon_x : 1 \to a \otimes a^*, \bar\epsilon_x : a^* \...

1
vote

0
answers

87
views

### Lagrangian subcategories of (non-pointed) braided tensor categories

I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\...

5
votes

1
answer

157
views

### Why is the category of strong braided functors from the braid category to a braided monoidal $M$ equivalent to the subcategory of *strict* functors?

This is my first, and probably my last, (for a while) posting on MO. I am very much a student and I don't claim to be a research mathematician, at all, but I have seen that sometimes "regular&...

7
votes

2
answers

182
views

### Where does the univeral $R$-matrix of $U_q(\mathfrak g)$ live?

Let $\mathfrak g$ be a complex simple Lie algebra and let $U_q(\mathfrak g)$ denote the Drinfeld-Jimbo quantum group associated to $\mathfrak g$. I will assume that $U_q(\mathfrak g)$ is a $\mathbb C(...

4
votes

1
answer

314
views

### Motivating quantum groups from knot invariants

Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...

9
votes

2
answers

343
views

### Coherence theorem in braided monoidal categories

In MacLane's Categories for the working mathematician, the author shows that the evaluation at 1 gives an equivalence of categories $\mathrm{hom}_{\mathrm{BMC}}(B,M)\simeq M_0$ where $B$ is the braid ...

3
votes

0
answers

125
views

### How to calculate the Lagrangian subgroup of $G\oplus\hat{G}$?

Let $G$ be an finite abelian group. We have known the following things:
Denote the Drinfeld center of $\operatorname{Rep}(G)$ by $\mathfrak{Z}_1(\operatorname{Rep}...

1
vote

1
answer

113
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 ...

5
votes

2
answers

244
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 ...

4
votes

1
answer

234
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 ...

4
votes

1
answer

123
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?

5
votes

1
answer

103
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 ...

5
votes

1
answer

438
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 $\otimes_{H}$ ?

5
votes

1
answer

406
views

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

$\newcommand\Mod[1]{#1\text{-Mod}}$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 ...

5
votes

2
answers

235
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 ...

4
votes

1
answer

210
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

0
answers

111
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 ...

7
votes

1
answer

322
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 ...

5
votes

0
answers

124
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 $\mathcal{B}$, ...

5
votes

1
answer

200
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 ...

12
votes

1
answer

276
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 ...

2
votes

1
answer

172
views

### Comodule Morita equivalence for Hopf algebras

Let $A$ and $B$ be two Hopf algebras, and denote by $\mathcal{M}^A$ and $\mathcal{M}^B$ their respective categories of right comodules. If we have a monoidal equivalence between $\mathcal{M}^A$ and $\...

10
votes

2
answers

666
views

### Is there a non-degenerate quadratic form on every finite abelian group?

Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, ...

4
votes

1
answer

368
views

### Does the functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ have adjoints?

Let $\mathcal{C}$ be a braided monoidal category. We have a canonical functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ from $\mathcal{C}$ to the Drinfeld center $\mathcal{Z}(\mathcal{C})$ sending an ...

1
vote

0
answers

96
views

### When is the action of the braid group on tensor powers of Yetter-Drinfeld modules faithful?

Let $V$ be a Yetter-Drinfeld module over a Hopf algebra $H$ with invertible antipode. Recall that $V$ is a braided vector space with braiding $\Psi\colon V\otimes V\to V\otimes V, v\otimes w\mapsto v_{...

2
votes

0
answers

72
views

### On $\Psi$-generating paths in the Bruhat order of a Weyl group

Let $W$ be a Weyl group with roots $R$ and positive roots $R^+$. Let $v\in W$ of length $r$. We call $\mathbb{m}=(\alpha_1,\ldots,\alpha_r)\in(R^+)^r$ a Bruhat path from $1$ to $v$ if $1\lessdot s_{\...

5
votes

2
answers

501
views

### Representation theory in braided monoidal categories

The crux of what I wish to know is what results from representation theory, a subject usually framed within the category $\text{Vect}_\mathbb{k}$, follow in more general braided monoidal categories? I ...

4
votes

0
answers

146
views

### additivity of trace with respect to short exact sequences

Let $\mathcal{C}$ be an abelian rigid symmetric monoidal category over a field $K$. Assume that the endomorphism ring of the tensor unit in $\mathcal{C}$ is $K$. If $X$ is an object in $\mathcal{C}$ ...

1
vote

0
answers

85
views

### Braided category inside braided 2-category

Let $\mathcal{C}$ be a semistrict braided monoidal $2$-category in the sense of [BN] (so in particular a strict $2$-category). Let $\mathcal{C}_1$ be the category of $1$-morphisms (objects) and $2$-...

8
votes

1
answer

245
views

### R-matrices and symmetric fusion categories

Given a $\mathbb{C}$-linear braided fusion category $\mathcal{C}$ containing a fusion rule of the form e.g.
\begin{equation}X\otimes Y\cong A\oplus B \oplus C\end{equation}
(where $A,B, C, X$ and $Y$ ...

3
votes

1
answer

127
views

### Integrals and finite dimensionality in braided Hopf algebras

Let $H$ be a Hopf algebra with invertible antipode. Let $A$ be a braided Hopf algebra in the Yetter-Drinfeld category ${}_H^H\mathcal{YD}$ over $H$.
A nonzero left integral in $A$ is a nonzero ...

4
votes

1
answer

153
views

### Is the center of an abelian rigid monoidal category, abelian?

Is the Drinfeld-Majid center of an abelian rigid monoidal category, abelian?
[stated in 1J of On the center of fusion categories" by Bruguières and Virelizier (link at Virelizier's page)]
In ...

11
votes

1
answer

676
views

### Is the category $\operatorname{sVect}$ an "algebraic closure" of $\operatorname{Vect}$?

$\DeclareMathOperator\sVect{sVect}\DeclareMathOperator\Vect{Vect}$The category $\sVect_k$ of (let's say finite-dimensional) super vector spaces can be obtained from the category $\Vect_k$ of (finite-...

1
vote

0
answers

84
views

### On reflexive bialgebras

Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...

4
votes

0
answers

93
views

### Tensor algebras in the bicategory $\mathsf{2Vect}$

To my knowledge there are two main approaches to categorify the notion of a vector space. I will refer to them as BC-2-vector spaces (Baez, Crans) and KV-2-vector spaces (Kapranov, Voevodsky). Both ...

3
votes

1
answer

481
views

### Examples of strict monoidal categories and monoidal categories with nontrivial associators

What are some "natural" motivating examples of the following:
i) A strict monoidal category,
ii) A monoidal with non-trivial associatots?
For i) the only examples I know are categories which ...

6
votes

1
answer

177
views

### Nonbraided rigid monoidal category where left and right duals coincide

In a braided rigid monoidal category $(\mathcal{M},\otimes)$ left and right duals coincide. What is an example of a rigid monoidal category where left and right duals coincide but there exist no ...

2
votes

0
answers

60
views

### Integrals in noncommutative graded algebras which are not necessarily Hopf

Let $\mathbf{k}$ be a field. Let $A$ be a finite dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbf{k}$-algebra such that $A^0=\mathbf{k}1$. Let $m$ be the maximal non-negative integer such that $A^m\...

4
votes

0
answers

71
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 ...

9
votes

4
answers

2k
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 ...

5
votes

1
answer

191
views

### Under what conditions is a symmetric tensor category equivalent to $\operatorname{\mathsf{Rep}}G$ for some group $G$?

Deligne's theorem on tensor categories states that for any symmetric tensor category $\mathcal{C}$ satisfying the subexponential growth condition, there is a fiber functor to $\mathsf{sVec}$ and that $...

4
votes

0
answers

98
views

### Scaling Yetter--Drinfeld Modules

A braided vector space is a pair $(V,\sigma)$ consisting of a vector space $V$, and a linear map $\sigma:V \otimes V \to V \otimes V$, satisfying the Yang--Baxter equation. Ee can scale the braiding ...

4
votes

0
answers

113
views

### semisimplicity of maps in braided vector spaces

Let $V$ be a finite dimensional braided vector space over $\mathbb{C}$.
This means that we have a map $$c_{V,V}:V\otimes V\to V\otimes V$$ which gives us an action of the braid group $B_n$ on $V^{\...

9
votes

2
answers

327
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 ...

17
votes

2
answers

3k
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

1
answer

156
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 ...