# Questions tagged [monoidal-categories]

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316
questions

**3**

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### 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 ...

**3**

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**1**answer

98 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 $\...

**1**

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106 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 ...

**4**

votes

**1**answer

324 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 ...

**2**

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**2**answers

69 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.

**3**

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147 views

### “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 (...

**5**

votes

**1**answer

127 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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91 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 ...

**5**

votes

**1**answer

232 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 ...

**8**

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**4**answers

660 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 ...

**7**

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**2**answers

206 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-...

**9**

votes

**1**answer

321 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{...

**2**

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**1**answer

121 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 ...

**5**

votes

**1**answer

155 views

### Categorical Description of Adjoint Representation

My apologies for a slightly vague question here. Let us fix a Lie algebra $\mathfrak{g}$ over a field $k$. Consider the symmetric monoidal category $Rep_{k}^{\otimes}(\mathfrak{g})$ of representations ...

**2**

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40 views

### Semisimplicity of the tensor identity in a multifusion category over an arbitary field

For a multifusion category $ \mathcal{C} $ over an algebraically closed field it is known that $ \text{End}(\mathcal{1}) $ is a commutative semi-simple algebra. See, for example, Theorem 4.3.1 in [1]. ...