# 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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### Monoidal categories in which some tensor products of morphisms are equal

I have come across a certain monoidal category $(C,\otimes,I)$ (let us say it is strictly monoidal to simplify the notations, as usual it does not matter very much) which satisfies the following ...
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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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### Is “monoidal category enriched over itself” the same as “closed monoidal category”?

If $M$ is a monoidal category, an enriched category over $M$ is a category $C$ whose hom-sets are viewed as objects in $M$. And a monoidal category $M$ is said to be closed if the tensor product ...
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### Can I construct a monoidal category on locally convex modules over algebras with approximate identiy

I am faced with the following problem: Let $A$ be a complete locally convex algebra with a uniform approximation of identity, that is a net $e_\lambda$, such that $p(e_\lambda a-a)\rightarrow 0$ for ...
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### Symmetric monoidal category which is not closed?

A monoidal category is symmetric if its tensor product is commutative up to natural isomorphism. And a symmetric monoidal category is closed if the tensor product functor has a right adjoint. We ...
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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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### Enriched category over non-monoidal category

I'm finally learning homological algebra, and the notion of an enriched category seems to be the right setting in which to define the $\mathbf{Ext}$ functor. Yet, the definitions of an enriched ...
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### For symmetric monoidal categories, the monoidal product is a monoidal functor

Has anyone got a reference for the following fact? If $\mathcal X$ is a symmetric monoidal category, then $\_\otimes\_\colon\mathcal X\times\mathcal X \to \mathcal X$ is a strong monoidal functor. ...
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### Definition of unitor in monoidal category

From https://ncatlab.org/nlab/show/monoidal+category, a monoidal category requires a natural isomorphism a natural isomorphism $\lambda: (1 \otimes (-)) \rightarrow ^\cong (-)$ with components ...
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### A monoidal category that preserves subobjects

Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be ...
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### Examples of weak monoidal Quillen equivalences

Schwede-Shipley introduced the notion of weak monoidal Quillen equivalences between monoidal model categories in "Equivalences of monoidal model categories". Are there any examples of such Quillen ...
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### Notation: why are categories enriched over $\mathcal V$?

In most references about enriched categories, $(\mathcal V, \otimes)$ is supposed to be a monoidal category and then $\mathcal V$-enriched categories are defined. Why is the letter $\mathcal V$ used ...
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### Existence and uniqueness of adjoints with respect to pairings

Let $V,W,L$ be $R$-modules over a commutative ring $R$. A pairing is an $R$-linear map $V\otimes W\to L$. An adjoint of an endomorphism $f:V\to V$ w.r.t a pairing $V\otimes W\overset{g}{\to}L$ is an ...
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### The category $\bf{FinVect}$ of finite vector spaces is rigid.

I am following Pavel Etingof et al's book on tensor categories. They give FinVect as an example of a rigid monoidal category, with evaluation map given by $\text {ev}_V(\epsilon\otimes v)=\epsilon(v)$...
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### A simple property of the $S$-matrix of a pre-modular category

I am following Pavel Etingof et al's book on tensor categories. In order to get used to the $S$-matrix of a pre-modular category and related concepts, I am trying to prove the following simple fact: ...
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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 ...
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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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### Is $\text{Aut}_{\text{br}}(C)$ braided?

Let $C$ be a braided monoidal category. The category $\text{Aut}_{\text{br}}(C)$ of braided monoidal autoequivalences of $C$ is monoidal with tensor product functor given by the composition $\circ$. ...
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### Constructing free monoid from free semigroup

Given a monoidal category $(\mathcal{C}, \otimes, I)$ with coproducts, the free monoid on an object $A \in \mathcal{C}$ is usually constructed by first constructing the free pointed object on $A$, i.e....
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### the Verlinde formula

The Verlinde formula writes the fusion coefficient in terms of S matrix. My question is that for fusion category without braiding, is there a similar formula which gives the fusion coefficient in ...
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### Equations on the free monoid??

While studying a bit of Joachim Lambek's calculus and some other applications of formal languages to the study of the structure of human language, I have come accross a reference to what authors like ...
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### Monoidal Equivalence: Two Definitions

One has the definition of a monoidal equivalence as in definition 12 of Baez's Some Definitions Everyone Should Know. I have also seen monoidal equivalence defined as a monoidal functor between ...
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### Why are the monoid objects in Mon(C) the commutative monoids?

Let $(C, \otimes, 1, \alpha, l, r)$ be a symmetric monoidal category. Can someone explain me why the monoids in the category of monoids Mon(C) are the abelian monoids ? Thanks for your help.
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### Tensoring of the identity morphism of the monoidal unit

This is a rather elementary question about strict monoidal categories. Let $C$ be a monoidal category, $I \xrightarrow{id_I} I$ be the identity morphism of the unit object. How do I show that for any ...
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### Modules can be viewed as monoid objects in some appropriate monoidal category?

I need to find a solution to have an inverse structure form: not classic modules over a monoid but monoids over modules. I had received this answer from here monoid objects are the minimal ...
Consider a symmetric monoidal category, and assume that it is closed, i.e. internal Homs exist. Recall that an object $X$ is dualizable if the canonical map $X \otimes DX \to \operatorname{Hom}(X,X)$ (...