# Questions tagged [monoidal-categories]

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353
questions

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### Automated rewriting of string diagrams in symmetric monoidal categories

Many algebraic structures, such as Frobenius algebras, or quasi-triangular Hopf algebras, can be formulated in an arbitrary symmetric monoidal category. They are given by a collection of morphisms, ...

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360 views

### Postnikov invariants of the Brauer 3-group

Given a commutative ring $k$ there is a bicategory with
algebras over $k$ as objects,
bimodules as morphisms,
bimodule homomorphisms as 2-morphisms.
This is a monoidal bicategory, since we can ...

**4**

votes

**1**answer

127 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 ...

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205 views

### Kan extensions between categories of monoid objects

Let $K\colon\mathcal{A}\longrightarrow\mathcal{B}$ be
a functor between $\mathcal{V}$-enriched categories. Endowing $\mathcal{A}$ and $\mathcal{B}$ with promonoidal
structures, we obtain induced ...

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61 views

### Oplax monoidal functors of $\infty$-categories

In Higher Algebra, a notion of lax symmetric monoidal functors (in what follows, I'll remove the adjective "symmetric", but I'm mainly interested in the symmetric situation) is defined : if you have ...

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**1**answer

122 views

### Infinitesimal categories and left duality

I have been reading Kassel's Quantum groups and there is something I can not understand.
In Section 4 of chapter $XX$, he introduces the notion of a Infinitesimal symmetric category, that is
a ...

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124 views

### Does Cantor Bernstein hold in a Closed Symmetric Monoidal Category?

In a closed symmetric monoidal category with $[I,X] \cong X$ for all $X$ is it true that having monomorphisms $m :A \rightarrow B$ and $m: B \rightarrow A$ is enough to imply $A \cong B$ ?
I tried to ...

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108 views

### A question about subobjects of the unit in a rigid abelian tensor category

I have a question about Proposition 1.17 in Deligne and Milne, Tannakian Categories (see here), in the last 4 lines of the proof.
I don't know how it follows from $U\otimes U\simeq U$ that $T=\ker(...

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104 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 ...

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108 views

### Sets of L-functions being “almost bimonoids”

Let $\mathcal{M}$ be a set of L-functions (where by L-function I mean any L-function associated to an automorphic representation of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ which is an element ...

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121 views

### Example of a projective bimodule with isomorphic left and right duals

What is an example of a non-free finitely generated $R$-bimodule $M$ satisfying
i) $M$ is projective as both a left and right $R$-module
ii) the right dual $\mathrm{Hom}_R(M,R)$ and the left dual ...

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123 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 ...

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**1**answer

159 views

### Rigidity for the category of comodules over a Hopf algebra

On this page
https://ncatlab.org/nlab/show/rigid+monoidal+category
there is a discussion of rigidity (left-right duality) for the catagory of
modules over a Hopf algebra. What happens if we look at ...

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112 views

### Monoidal structure in which oplax monoidal functors correspond to monoid objects

$$
\def\cat#1{{\mathbf{#1}}}
\def\opcat#1{{\mathbf{#1}^{\mathrm{op}}}}
$$
A symmetric lax monoidal functor $F : \cat{C} \rightarrow \cat{D}$ between monoidal categories $(\cat{C}, \otimes, I)$, $(\cat{...

**4**

votes

**1**answer

211 views

### Existence of a multiplication bifunctor for the category of groups

For $\mathsf{Grp}$ the category of groups, a bifunctor $M: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is a multiplication bifunctor if:
$M(C_n,C_m) \simeq C_{nm}$,
$M(C_1,G) \simeq M(G,C_1) \...

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**1**answer

213 views

### Existence of an addition bifunctor for the category of groups

Let $\mathsf{Grp}$ be the category of groups. A bifunctor $A: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is an addition bifunctor if:
$A(C_n,C_m) \simeq C_{n+m}$,
$A(C_0,G) \simeq A(G,C_0) \...

**12**

votes

**1**answer

504 views

### Classifying space for Thompson's group F?

Let $\mathcal C$ be the free monoidal category generated by an object $X$, and a morphism $X \otimes X \to X$.
This category contains exactly two connected components: that of the monoidal unit $1\in ...

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votes

**2**answers

232 views

### The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors

In 6.5 of the book by Kelly,
Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005.
the author claims that the $2$-category $\mathsf{Cat}_{\...

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**1**answer

232 views

### Existence of a finite group with a given decomposition for a tensor square of one irreducible complex representation

In this post, irrep and dim mean "irreducible complex representation" and "dimension", respectively.
It would be helpful (in a problem of monoidal category) to find a finite group $G$ with (at least) ...

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**1**answer

222 views

### Monoidal categories from the projective modules of a ring

Let $R$ be a not necessarily commutative ring, and denote by $_R\mathrm{lp}_R$ the category of $R$-bimodules, which are finitely generated projective as left modules, with morphism $R$-bimodule maps, ...

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100 views

### Braided monoidal categories

I know that it has been shown that $E_2$ algebra objects in Categories are simply braided monoidal categories.
In particular, Lurie says that an $E_2$-monoidal structure on the infinity-category $N(C)$...

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107 views

### A semicartesian monoidal category with diagonals is cartesian: proof?

The nLab states that a semicartesian monoidal category equipped with natural transformations $\delta_x : x \to x \otimes x$ such that $\pi_1 \circ \delta_x = 1_x$ and $\pi_2 \circ \delta_x = 1_x$ (...

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106 views

### Generalization of monoidal category with tensor products of $n$ objects

I'm looking for a generalization of monoidal categories, say $n$-monoidal categories, s.t. an ordinary monoidal category is the $n=2$ case. For general $n$, naively it should consist (among other data)...

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50 views

### Monoidal V-categories, and monoids

I am guessing that the definition of monoidal V-category is a V-category $\mathbf{A}$ together with a V-functor $(\boxtimes) \colon \mathbf{A} \times \mathbf{A} \to \mathbf{A}$ and a functor $i \colon ...

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130 views

### Internal logic in topos theory, monoidal categories, and quantum mechanics

To obtain the internal logic of a topos (roughly speaking), we associate a type of free variable with an object, and a statement about such a variable with a subobject of that object. Intuitively, the ...

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116 views

### Projective limits of monoidal categories

Increasingly harder question, but a reference for the first would be ok:
Is the category of (symmetric?) monoidal categories closed for limits like products?
Is it true that the underlying category ...

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538 views

### Monoidal category that is not spacial

Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom
This diagram says that for ...

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111 views

### Internal commutative monoid gives commutative monad

Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object.
The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...

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239 views

### Non-linear Galois descent

This question is about Galois theory. So let $K / k$ be a Galois extension of fields. Let us assume that $K / k$ is finite dimensional, though everything can be made to work in the profinite case by ...

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758 views

### Unitors and projections in cartesian category

In a cartesian monoidal category we have the product with two projections $\pi_1$ and $\pi_2$, and the terminal object $1$. We also have unitors $\rho_A \colon A \times 1 \to A$ and $\lambda_A \colon ...

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### A question about the Tannaka-Krein reconstruction of finite groups

In Chapter 15 Section 15.2.1 of Quantum Groups and Noncommutative Geometry, 2nd edition, the authors raised a question: can we reconstruct a finite group $G$ from its category of finite dimensional ...

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123 views

### Closed embeddings of monoidal categories in *-autonomous ones

It's often very convenient for objects in a monoidal category to have duals. Hence, it's natural to wonder whether an arbitrary monoidal category can be embedded in one where all objects have duals. ...

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392 views

### Tannaka-Krein duality in Chari-Pressley's book

I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here.
V.Chari and A.N.Pressley in their "Guide to Quantum ...

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108 views

### Is there an example of two strict monoidal categories which are (monoidally) equivalent, but not strictly?

By strict equivalence, I mean a monoidal equivalence whose underlying monoidal functors are strict, and here I am looking for two monoidal categories which are not strictly equivalent.

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**1**answer

271 views

### Krein's theorem in the Tannaka-Krein duality

In the Tannaka-Krein duality theory, the Krein theorem describes the conditions under which a given category $\varPi$ is a category of finite-dimensional representations of some compact group $G$:
...

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55 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 ...

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### Monoidal category of irreducible highest weight modules of the Virasoro algebra

I'm trying to see if I can construct a monoidal category $\mathbf{C}$ whose objects are the irreducible unitary highest weight representations of the Virasoro algebra.
I am thinking on doing the ...

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174 views

### What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?

A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...

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**1**answer

297 views

### A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules

Does anyone have a proof for the following Lemma?
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\...

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132 views

### The relationship between representations of groups and evaluation and coevaluation maps for $vect_{G}$ module categories

Let $G$ be a finite group and $vec_{G}$ be the monoidal category of finite dimensional $G$-graded vector spaces.
Given any $vec_{G}$ module category $\mathcal{M}$ we can define a dual module category ...

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**1**answer

338 views

### Lifting ring homomorphism of Grothendieck rings to functor of semisimple categories

I have two $\mathbb{C}$-linear semisimple tensor categories $C$ and $D$. Let $K(C)$ and $K(D)$ be their Grothendieck groups. I have a specific ring homomorphism $f \colon K(C) \to K(D)$ that I would ...

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111 views

### Example of non-symmetric biclosed monoidal category

Does there exists any example of non-symmetric biclosed monoidal category ? By biclosed, I mean right-closed and left-closed.

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### “Fundamental theorem for Hopf modules”

I am studying Hopf algebras in categories, and I hope, somebody could help me with the following.
Joost Vercruysse in his paper Hopf algebras---Variant notions and reconstruction theorems writes (...

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**1**answer

144 views

### monoidality of $ A\otimes (-) $ with $ A $ monoid belonging to the center

Let $(\mathcal{C}, \otimes)$ a monoidal category, and $(A, m, e)$ a monoid (where $m: A\otimes A\to A$, $e: I\to A$ ecc. ), with $(A, u)$ belonging to the centre of $(\mathcal{C}, \otimes)$: $u: A\...

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176 views

### Tensor schemes “with relations”

In "The geometry of tensor calculus I," Joyal and Street introduce the notion of a tensor scheme, a set of abstract objects, together with formal morphisms between words in this set. They then prove ...

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247 views

### Tannaka duality for closed monoidal categories

I asked this some time ago at mathstackexchange, and people there explained to me the mathematical part of what I was asking, but the question about references remains open. In my impression, people ...

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835 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 ...

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217 views

### Multiplication and division by a morphism under the “inner composition” in closed monoidal categories

I asked this a week ago at math.stackexchange, without success, so I hope it will be appropriate here.
Let ${\mathcal C}$ be a symmetric closed monoidal category, and let me denote the internal hom-...

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348 views

### Trace in the category of propositional statements

By the result in this paper, there exists a categorification of the trace of a linear operator that generalizes to any endomorphism of a dualizable object in a symmetric monoidal category ($\textbf{...

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147 views

### On the Group Structure of Morphism Set of a Strict 2-Group

The standard definition of a strict 2-group says that it is a strict monoidal category in which every morphism is invertible and each object has a strict inverse.
Also it is a well known fact that a ...