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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 ...
      3
      votes
      1answer
      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
      vote
      0answers
      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
      1answer
      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
      votes
      2answers
      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
      votes
      0answers
      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
      1answer
      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\...
      1
      vote
      0answers
      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
      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 ...
      8
      votes
      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 ...
      7
      votes
      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-...
      9
      votes
      1answer
      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
      votes
      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 ...
      5
      votes
      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 ...
      2
      votes
      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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