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Questions tagged [monoidal-categories]

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor ⊗ : C × C → C which is associative up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism. (Def: http://en.m.wikipedia.org/wiki/Monoidal_category)

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Uniqueness of Duals in a Monoidal Category

Given a monoidal category ${\cal C}$, and $X \in {\cal C}$, we define a left dual of $X$ to be an object $X^*$ such that there exist morphisms $\epsilon:X^* \otimes X \to I$, and $\eta:I \to X \otimes ...
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Why are duals in a rigid/autonomous category unique up to unique isomorphism?

I'm having trouble understanding the following statement: "In a rigid category, duals are unique up to unique isomorphism." It seems to me that this isomorphism is not unique. Let me try to give a ...
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Understanding associators as natural transformations

Reading Baez and Stay's "Rosetta Stone," and trying to understand the definition of monoidal category on page 12, I read that a monoidal category requires a natural isomorphism called the associator, ...
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Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
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Monoid in the category of endofunctors and Monoid as a category with one object

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...
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A conceptual understanding of transmutations (and bosonizations) of (braided) Hopf algebras

Consider a coquasitriangular Hopf-algebra $(H,\mu,\eta,\Delta,\epsilon, S)$ over a field $\mathbb F$ with characteristic zero and the braided monoidal category $\mathcal C$ of $H$-right-comodules. We ...
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Tensor products from internal hom?

Monoidal categories come with tensor products, and sometimes, these categories are biclosed, i.e each restriction of the tensor bifunctor has a right adjoint. If the category happens to be symmetric, ...
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Associativity of Day convolution

I'm trying to follow Day's argument to prove that $[\mathbf C,\mathbf{Sets}]$, where $\bf C$ is symmetric monoidal, is itself symmetric monoidal, but I'm stuck at the very beginning. Is there a way to ...
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Reasons for coherence for bi/monoidal categories

Here by coherence conditions I mean those axioms imposed on associators and unities that grant that the groupoid generated by such morphisms is a poset, i.e. any two parallels morphisms in this ...
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Does an adjoint of the Hom functor make a category monoidal?

In the category of modules, the tensor product functor is the left adjoint of the covariant Hom functor. Similarly in the category of sets, the Cartesian product functor is the left adjoint of the ...
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Morphisms in a symmetric monoidal closed category.

Let $\mathcal C$ be a symmetric monoidal closed category. This means that every functor $- \otimes B$ has a right adjoint $[B, -]$. Let $I$ be the unit and let $\rho \colon - \otimes I \to 1_{\mathcal ...
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Detecting skew-primitives in representation categories

Suppose $H$ and $H'$ are two (possibly infinity dimensional) Hopf algebras which are not isomorphic as Hopf algebras, but are isomorphic as algebras. More specifically they are not isomorphic as Hopf ...
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At a closed monoidal category, how can I derive a morphism $C^A\times C^B\to C^{A+B}$?

Let $A$, $B$ and $C$ be objects of a closed monoidal category which is also bicartesian closed. How can I derive a morphism $C^A\times C^B\to C^{A+B}$? $(-)\times (-)$ denotes the product, $(-)+(-)$ ...
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Adjoint functors on categories of monoids induced by a strong monoidal adjunction

Let $C$ and $D$ be monoidal categories and let $F:C\rightarrow D:G$ be a strong monoidal adjunction (adjunction in which $F$ and $G$ are strong monoidal functors). There are induced functors $Mon(F):...
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Is every equivalence of monoidal categories a monoidal equivalence?

Let $C$ and $D$ be monoidal categories. Let $T:C \to D$ be a functor that gives the equivalence of $C$ and $D$ as just categories. My question is whether such an equivalence $T$ is always a monoidal ...
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What is an abelian group in a symmetric monoidal category?

Reading Tom Leinster's book Higher operads, higher categories , I found Example 2.1.5 a bit puzzling. It says: For instance, let $V$ be a symmetric monoidal category and let $Ab(V)$ be the category ...
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Endomorphisms in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category generated by one element $X$ such that $End(X)=G$ where $G$ is a finite group. Is it true that, for any object $A \in \mathcal{C}$, $End(A)$ is ...
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Monoidal equivalence is: A monoidal functor that is also an equivalence of categories

Let $(\mathcal C, \otimes, I)$ and $(\mathcal D, \otimes', I')$ be two monoidal categories. And let $(F,\phi, \psi)$ and $(G,\phi', \psi')$ be a monoidal functors from $\mathcal C$ to $\mathcal D$. ...
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The abstract definition of commutative monoids

In trying to begin to understand the idea of a $k$-tuply monoidal $n$-category, I'm already a bit stuck on the idea (Baez, nLab) that a commutative monoid can be defined as a monoid object in the ...
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Every monoidal category is strictly equivalent to a strict monoidal category.

I am reading Joyal and Street's article "braided tensor categories". The following theorem is proved (Theorem 1.2). Let $ C $ be a category. Let $FC$ and $F_sC$ be respectively the free monoidal ...
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Categorical approach to form of simple representation of product of groups

Throughout, $\mathsf C_1,\mathsf C_2$ are small categories and $\mathscr V$ is a bicomplete symmetric monoidal closed category. I'm trying to understand some of the basic theory of group ...
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Does an adjoint of an internal Hom functor of a prounital closed category define a tensor product?

A closed category is a category equipped with internal Hom functors along with a unit object. Now this answer shows that if $C$ is a closed category whose internal Hom functor has a left adjoint, ...
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Free monoidal category over a set

From nLab's article on coherence theorems, there seems to be a notion of free monoidal category over a set $S$. I guess this corresponds to the left adjoint to the functor $Ob : MonCat \to Set$ which ...
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Elements of the Monoid in the category of endofunctors

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...