All Questions
Tagged with braided-tensor-categories ct.category-theory
60
questions
18
votes
2
answers
4k
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 ...
- 815
13
votes
1
answer
589
views
Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory
Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue ...
- 25.1k
12
votes
2
answers
691
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 ...
- 5,485
12
votes
1
answer
509
views
Is there a "killing" lemma for G-crossed braided fusion categories?
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 ...
- 5,485
11
votes
1
answer
694
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-...
- 59k
10
votes
4
answers
1k
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 ...
- 1,852
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 ...
- 815
9
votes
2
answers
593
views
Why is a braided left autonomous category also right autonomous?
In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that:
Lemma 4.17 ([23, Prop. 7.2]). A braided ...
- 345
9
votes
4
answers
930
views
The dual of a dual in a rigid tensor category
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.
- 953
9
votes
2
answers
351
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 ...
- 5,485
9
votes
2
answers
360
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 ...
- 201
8
votes
1
answer
917
views
Twists, balances, and ribbons in pivotal braided tensor categories
Let $\mathcal{C}$ be a pivotal tensor category. Feel free to assume finiteness, semisimplicity, fusion, sphericality, unitarity or whatever makes things interesting. Which of the following structures ...
- 1,189
8
votes
1
answer
279
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 $Y$ ...
- 277
8
votes
1
answer
298
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^* \...
- 779
8
votes
1
answer
548
views
Morita equivalent algebras in a fusion category
Let $\mathcal{C}$ be a braided $\mathbb{k}$-linear fusion category ($\mathbb{k}$ algebraically closed; if necessary to answer my question you can also assume $\mathcal{C}$ to be pivotal or even ...
- 295
8
votes
0
answers
374
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 ...
- 5,485
7
votes
3
answers
585
views
Does one of the hexagon identities imply the other one?
Suppose we have a monoidal category equipped with additional data that almost makes it a braided monoidal category except that only one of the hexagon identities
is satisfied.
Can we then prove the ...
- 35.6k
7
votes
3
answers
883
views
Nerves of (braided or symmetric) monoidal categories
I'm looking for references on the structure which can be roughtly described as follows: given a (braided or symmetric) monoidal category $C$, I want to consider a simplicial set $N(\mathbf{B}C)$ with ...
- 6,679
7
votes
2
answers
875
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 ...
- 953
7
votes
1
answer
334
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 ...
- 261
7
votes
1
answer
368
views
Bimodule categories realized as internal bimodules
Let $\mathcal C$ be a finite tensor category, and $\mathcal M$ a finite left $\mathcal C$-module category. By a result of P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik (http://www-math.mit.edu/~...
- 295
6
votes
1
answer
338
views
Name for an Isomorphism in a Monoidal Category that Satisfies the Braid Relation
Let $({\cal C},\otimes)$ be a monoidal category, $X$ an object in ${\cal C}$, and $\Psi:X \otimes X \to X \otimes X$ an isomorphism such that $\Psi$ satisfies the braid relation:
$$
(\Psi \otimes \...
6
votes
2
answers
758
views
Module categories over symmetric/braided monoidal categories
Given an algebraically closed field $k$ and a finitely generated commutative $k$-algebra $A$, all simple modules over $A$ are 1-dimensional
What is the analogous statement for symmetric monoidal $k$-...
- 1,358
6
votes
1
answer
179
views
Nonbraided rigid monoidal category where left and right duals coincide
In a braided rigid monoidal category $(\mathcal{M},\otimes)$ left and right duals coincide. What is an example of a rigid monoidal category where left and right duals coincide but there exist no ...
6
votes
1
answer
327
views
When is this braiding not a symmetry?
Given a topological space $X$ instead of forming the fundamental groupoid $\pi(X)$ which is the category whose objects are the points and morphisms the homotopy classes of paths one can also form the ...
- 1,614
6
votes
2
answers
453
views
Does the dual of an object with trivial symmetry also have trivial symmetry?
Let $C$ be a symmetric monoidal category. I am interested in objects $X \in C$ such that the symmetry
$S_{X,X} : X \otimes X \cong X \otimes X$
is equal to the identity. There are many examples of ...
5
votes
2
answers
1k
views
When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular?
Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural ...
- 5,485
5
votes
2
answers
247
views
What is the proof of the compatibility of a braiding with the unitors?
I am specifically referencing the property that, given a braided monoidal category with a braiding $c$ and left and right unitors $\lambda, \rho$,
$$
\lambda_A \circ c_{A,I}=\rho_{A},
$$
for any ...
- 303
5
votes
2
answers
235
views
Constructing the inverse of a braiding in a braided pivotal category
Assume we have a braided pivotal monoidal category. This means we assume the braiding $c$ to be a natural isomorphism. But looking at the corresponding string diagram, it seems to me as if we could ...
- 105
5
votes
1
answer
194
views
Under what conditions is a symmetric tensor category equivalent to $\operatorname{\mathsf{Rep}}G$ for some group $G$?
Deligne's theorem on tensor categories states that for any symmetric tensor category $\mathcal{C}$ satisfying the subexponential growth condition, there is a fiber functor to $\mathsf{sVec}$ and that $...
- 275
5
votes
1
answer
253
views
On the existence of a square root for a modular tensor category
The center $Z(\mathcal{C})$ of a spherical fusion category $\mathcal{C}$ (over $\mathbb{C}$) is a modular tensor category.
Question: What about the converse, i.e., can we characterize every modular ...
- 25.5k
5
votes
1
answer
439
views
Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$
$\newcommand\Mod[1]{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between
$$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$
The proof in "Tensor Categories ...
- 81
5
votes
1
answer
167
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&...
- 477
5
votes
1
answer
289
views
Is every premodular category the *full* subcategory of a modular category?
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 ...
- 5,485
5
votes
1
answer
477
views
Explicit description of a free braided monoidal groupoid with inverses
Let G be a braided monoidal groupoid: it does no harm to suppose that the monoidal product on G is strictly associative, so I'll do that.
"With inverses" means that for every object $X$ of G, there ...
- 26.5k
5
votes
1
answer
446
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 $\otimes_{H}$ ?
- 69
5
votes
2
answers
507
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 ...
- 191
5
votes
0
answers
128
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 $\mathcal{B}$, ...
- 52.2k
4
votes
1
answer
422
views
When modular tensor categories are equivalent?
I asked this question at math stack exchange math stack exchange but I haven't got any answer yet there.
I would like to know when we say that two modular tensor categories are equivalent.
Is it ...
user46846
4
votes
1
answer
383
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 ...
- 1,527
4
votes
1
answer
230
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 ...
- 569
4
votes
1
answer
474
views
Faithful exact functors to tensor categories
Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P $ an additive $k$-linear exact and faithful functor. I want to ...
- 4,454
4
votes
1
answer
236
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 ...
- 303
4
votes
1
answer
155
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 ...
- 91
4
votes
1
answer
427
views
About a categorical definition of graded (coloured) algebra
The definition of graded algebra had a growing interest in algebra and mathematical physics (see $[GTC]$), I see that this topic has an elegant and simple categorical generalization, but I have not ...
- 4,095
4
votes
0
answers
99
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 ...
- 1,527
4
votes
0
answers
74
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 ...
- 1,527
4
votes
0
answers
367
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 - \...
- 5,485
3
votes
3
answers
907
views
When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?
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 ...
- 5,485
3
votes
1
answer
535
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 ...
- 1,104