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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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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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Lurie's reformulation of symmetric monoidal tensor categories in HA

In the introduction to Chapter 2 of Jacob Lurie's Higher Algebra (entitled "$\infty$-Operads"), a category $\mathcal{C}^\otimes$ is constructed from an arbitrary symmetric monoidal category $\mathcal{...
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Every under category with pushouts is monoidal

Maybe a stupid question, but I can't figure it out. Suppose we have some category $C$ where all pushouts exist, $c\in Ob C$ and $c\downarrow C$ an under category (i.e. a category with objects of the ...
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Dualizable presheaves with respect to Day convolution

Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, Set)$ its category of presheaves regarded as a closed symmetric monoidal category via Day ...
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Intuition of factorization systems' role in the construction of free algebras

Kelly's article uses factorization systems for constructing free monoids. What's the intuitive reading of factorization systems in this case? Also, for example, Barr in this article cites a result by ...
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Precondition “small category” in functor category

I am currently working on a exercise sheet about categories. There are two exercises: In the first parts I have to show that the vertical composition and the horizontal composition of two natural ...
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Equality of maps in a monoidal category

Let $C_1,C_2$ be monoidal categories (aka tensor categories) with tensor bifunctor $$\otimes_i: C_i\times C_i\to C_i$$ and tensor units $1_i$. Assume I have a monoidal functor $F:C_1\to C_2$, it ...
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Commuting diagram for (monoidal) functors

Assume we have two categories $\cal A,B$ and four functors $F,G:\cal A\to B$, $D:\cal A\to A$, $E:\cal B\to B$. In order to show an equality like $$E\circ F=G\circ D\quad (1)$$ I have to prove that ...
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Definition of monoidal categories in Etingof’s “Tensor Categories”

I am quite confused on page 22,23 of the unit axiom of a monoidal category. Unit axiom: $L_1:X \mapsto 1 \otimes X$ and $R_1:X \mapsto X \otimes 1$ are autoequivalences of $C$. (end of pg 22)...
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$(\mathrm{Vect}_k,\otimes_k)$ as a non-strict monoidal category

What is the easiest way to see (or understand) that the category of vector spaces over a field $k$, endowed with its usual modoidal structure $\otimes_k$, is not a strict monoidal category.
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Unit isomorphism in SVECT

What is the unit isomorphism $$X\otimes\mathbb{C}^{1|0}\cong X$$ in the monoidal caregory of super-vector spaces? Is it $$x\otimes\lambda\mapsto \lambda x$$ like in the monoidal category of vector ...
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Adjunction of functors between monoidal categories induces adjunction between their categories of internal monoids

Here's a similar question, but I'm not sure if what I'll ask is already answered there. Let $(C,\odot), (D,\square)$ be monoidal categories. If $F \colon D \to C$ is left adjoint to $U \colon C \to D$...
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Affine space as a ringed space; is this the correct definition?

Let $k$ denote a field and let $\mathbb{A}^n$ denote affine $n$-space. Then if I understand correctly, it's best to view $\mathbb{A}^n$ as a ringed space. I'm a bit unsure as to what the structure ...
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Group homomorphism yields to monoidal functor

Consider a group $G$. We can get to a monoidal category $\cal G$ with objects being the group elements, tensor product given by group multiplication and just the identity morphisms as morphisms. ...
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Finding the Drinfeld centre of a category

I have the following unitary monoidal spherical category C: Simple objects: $\{1,x,y\}$. Non-trivial Fusion Rules: $$x\otimes y=x=y\otimes x$$ $$x\otimes x=1 \oplus 2x \oplus y$$ I would like to ...
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Is the class of dualizable objects in an abelian monoidal category closed under sums, kernels and cokernels?

Goodmorning to everybody. I am in the following situation. I have been told that in an abelian monoidal category (I assume this means an abelian category $\mathscr{A}$ with a monoidal structure $(\...
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In a tensor category, does $X\otimes Y\cong 0$ imply $Y\cong 0$ for non-zero $X$?

By a tensor category I mean a locally finite rigid $k$-linear abelian category with bilinear tensor product, and such that $\operatorname{Hom}(1,1)\cong k$.$^1$ Suppose we fix some non-zero object $...
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When is a monoidal structure on $\mathrm{Mod}_A$ induced by a bialgebra structure on $A$?

Fix a field $k$, and let $A$ be a (commutative, coassociative, counital) $k$-bialgebra. Write $\otimes = \otimes_k$. The category $\mathrm{Mod}_A$ of $A$-modules admits the structure of a monoidal ...
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Monoids as Categories With One Object of Specific Type

I was reading David Spivak's excellent book on category theory, where he uses a category to define a monoid. Another text by Tom Leinster defines monoids in the same way (I've only read through the "...
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Smallest abelian braided monoidal subcategory containing an object $V$

Let $\mathcal{C}$ be an abelian braided monoidal category with countable direct sums compatible with the tensor product (i.e. $X\otimes \bigoplus_{i \in \mathbb{N}} V_i \cong \bigoplus_{i \in \mathbb{...
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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 ...
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How to see that endotransformations of fiber functor have a coalgebra structure?

This question is based on section 5.2 in Tensor Categories, by Etingof et al. Note also that the question is pretty much in the title and what follows is just some background along with my fruitless ...
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A well-known property of dualizable objects in a monoidal category

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 $X \otimes DX \to \operatorname{Hom}(X,X)$ (...
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Examples of weakly dualizable objects in a non-closed monoidal category.

The following is a straightforward generalization of the notion of dualizable object in a symmetric monoidal category given in Duality, Trace and Transfer by Albrecht Dold and Dieter Puppe to non-...
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What is a Presheaf-category enriched pullback?

I have a question about presheaf-enriched categories, like sSet for example that I think is pretty basic, but I don't know how to go about. So I have a category $C$, like $\Delta^\text{op}$, that is ...
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Division (or internal hom) notation in monoidal categories

At the end of section A.1.4 of the book "higher topos theory," there is a formula $X\otimes (C\otimes D)\simeq (X\otimes C)\otimes D$ which means the action property of tensoring in enriched ...
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Question concerning functors which are equivalences

I am currently reading about monoidal categories in the book "Tensor categories" and I am confused about the following: The authors write that the functors $L_1: X \mapsto 1 \otimes X$ and $R_1: X \...
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About the internal hom in a symmetric monoidal closed category

Let $\mathcal{C}$ be a symmetric monoidal closed category. My question is the following: Given three objects $X$, $Y$ and $Z$, and a morphism $f \colon Y \to Z$ in $\mathcal{C}$, does it ...
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Show that the unit object in a monoidal category is unique up to isomorphism.

Let $(C, \otimes, 1, \alpha, l, r)$ be a monoidal category. I would like to prove that the unit object is unique up to isomorphism. More precisely, I need to show that for any triples $(1, l, r)$...
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Let $(C, \otimes, \alpha, 1, l, r)$ be a monoidal category. Show that $l_{1 \otimes A} = id_1 \otimes l_A$ and $r_{A \otimes 1} = r_A \otimes id_1$.

Let $(C, \otimes, \alpha, 1, l, r)$ be a monoidal category. I would like to show that for any object $A$ in $C$ the equalities $l_{1 \otimes A} = id_1 \otimes l_A$ and $r_{A \otimes 1} = r_A \...
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Understanding The Equations Behind Dual Objects In Monoidal Categories

I think about the identities in monoidal categories $C$ (triangle and pentagon) the following way: Let $\Sigma=(\otimes,1)$ denote the monoid signature, $T(\Sigma,C_0)$ the set of terms over $\Sigma$ ...
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Showing that naturality of this transformation follows from dinaturality of elements.

This is from Lemma 3.7 in $SL(2,\mathbb{Z})$-action for ribbon quasi-Hopf algebras. Let $\mathcal{C}$ be a braided monoidal category with left duals (small, strict), define the functor $F=-^*\otimes-...
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Are there “distributive bicategories”?

In the bicategory $\mathsf{Bimod}$ of rings, bimodules and bimodule morphisms there is also a direct sum making every category $\mathsf{Bimod}(\mathsf{R},\mathsf{S})$ of $(\mathsf{R},\mathsf{S})$-...
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Free monoidal category over a set II

In a previous question, a description of the free monoidal category over a set was given. Basically, it consists of formal expressions as objects and morphisms generated by associators and unitors, ...
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How to represent convolution matrix as a string diagram?

I'm trying to represent common machine learning/deep learning concepts and operations in the form of string diagrams. The many categorical quantum mechanics papers by Coecke et al have given me a good ...
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Lax/colax monoidal adjunction in terms of unit and counit

Following nLab's page on monoidal adjunctions, it seems there is a bunch of options for an adjunction to become "monoidal" in some sense. I think that the one called monoidal adjunction in the nLab ...
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The bicategory of monoidal categories vs the tricategory of one-object bicategories.

A monoidal category is a bicategory with one object. So from this perspective, one would be tempted to define a tricategorie $ \overline{\mathbf{Mon Cat}} $ of monoidal categories with lax functors ...
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Difference between monoidal and tensor categories

Is a monoidal category just another word for tensor category or are those two different (but still similiar) things in the sense that one of them is more general? Are those categories supposed to be ...
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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 ...