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# Questions tagged [monoidal-categories]

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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 ...
1answer
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### 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 ...
1answer
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 ...
4answers
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 ...
2answers
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-...
1answer
321 views

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 ($\textbf{... 1answer 121 views ### On the Group Structure of Morphism Set of a Strict 2-Group The standard definition of a strict 2-group says that it is a strict monoidal category in which every morphism is invertible and each object has a strict inverse. Also it is a well known fact that a ... 1answer 155 views ### Categorical Description of Adjoint Representation My apologies for a slightly vague question here. Let us fix a Lie algebra$\mathfrak{g}$over a field$k$. Consider the symmetric monoidal category$Rep_{k}^{\otimes}(\mathfrak{g})$of representations ... 0answers 40 views ### Semisimplicity of the tensor identity in a multifusion category over an arbitary field For a multifusion category$ \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]. ...

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