Questions tagged [monoidal-categories]

A monoidal category, also called a tensor category, is a category $\mathcal{C}$ equipped with a bifunctor $\otimes\colon \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ which is associative up to a natural isomorphism, and an object $\mathbb{1}$ which is both a left and right identity for $\otimes$ up to a natural isomorphism.

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Further interesting examples? Obtaining (co)monoids from dual objects

1. Context Obtaining (co)monoids from dual objects Let $(C, \otimes, I, a, l,r)$ be a monoidal category. To simplify notation (and work with string diagrams) we assume that $C$ is strict. Let $V \in ...
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Module structure on the dual: string diagram in $vect_{\mathbb k}$.

1. Context Let $H$ be a Hopf algebra over a field $\mathbb k$. Let $(V, p)$ be a finite dimensional (left) $H$-module. We want to endow its dual vector space $V^*$ with the structure of a (left) $H$-...
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Rank of $R^n$ in characteristic $n$

I'm reading Deligne-Milne's introduction to Tannakian categories, and I noticed a troubling consequence of the definition of rank in a rigid ACU tensor category $(\mathcal{C}, \otimes)$. Specifically, ...
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Monoidal category: Inequivalent associators

Context In my lecture notes on tensor categories it says: "For a given category $C$ and a given tensor product $\otimes$, inequivalent associators can exist." Questions What notion of ...
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Why are these two functors adjoint pairs?

In the Braided Tensor Categories paper of 1993, Joyal and Street make a nontrivial claim with no proof. It is critical to their work and I can't figure out why it's true. Let $\mathbb{P}$ be the ...
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Braided ribbon Hopf algebra

I am interested in an extension of S. Majid's notion of braided quasi-triangular Hopf algebras in braided monoidal categories to braided ribbon Hopf algebras in such categories. Let me explain what I ...
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Which is an example of a monoidal category which cannot be braided?

This is probably a most stupid question, but I really do not have a profound knowledge of monoidal and monoidal braided categories, I only skimmed across them in a course on Hopf Algebras. My question ...
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The “symmetric” property of Day convolution.

This question has to be divided into the following parts: The definition of Day convolution in nlab To define Day convolution, it assumes that $V$ be a closed symmetric monoidal category with all ...
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How does Mac Lane's coherence theorem follow from the fact that a tensor category is equivalent with a strict tensor category?

I'm reading a book and it proves the following theorem: Given a tensor category $(\mathcal{C}, \otimes, I, a, l,r)$, there exists a strict tensor category $\mathcal{C}^{str}$ (that is, the ...
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Dual of Hom spaces in linear categories

Suppose you have a monoidal $\mathbb{C}$-linear category $\mathcal{C}$ It is stated in nlab (See here https://ncatlab.org/nlab/show/semisimple+category#:~:text=Definition,of%20finitely%20many%20simple%...
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Variants of rigid monoidal category for infinite dimensional spaces

It is well-known that the notion of a rigid monoidal category does not interact well with infinite-dimensional spaces. To be more precise, the category of all vector spaces is not rigid, see for ...
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Projective cover of simple counts occurences of simple in composition series?

On page 11 in Tensor Categories is the statement Let $\mathcal{C}$ be a finite abelian $k$-linear category. Then for any $X,Y \in \mathcal{C}$ with $X$ simple we have \begin{align*} \dim_k \...
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Additivization of functors in an abelian monoidal category

Crossposted on MathOverflow here. I'm having trouble with the proof of Lemma 2.9 in "Cohomology of Monoids in Monoidal Categories" by Baues, Jibladze, and Tonks, and I was wondering if someone could ...
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Natural isomorphism in linearly distributive categories with left and right dualities

Let $\mathrm{C}$ be a linearly distributive category with a left and right duality, i.e. it is a monodical category "twice": once for the bifunctor $\otimes:\mathrm{C}\times\mathrm{C}\to\mathrm{C}$ ...
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Checking monoidal structure axiom on simple objects

If we consider a semisimple linear monoidal category $\mathcal{C}$, $\mathcal{D}$ another linear monoidal category, a linear functor $F:\mathcal{C}\rightarrow \mathcal{D}$ and natural isomorphisms $J_{...
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Monoidal structure preserving order

I am working with a category $\mathcal{C}$ in which the hom-sets are orders. Alternatively, we could look at it as a bicategory in which hom-sets for 2-cells are thin. Is there a notion of monoidal ...
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What is a module over an algebra in the category theoretic sense?

Can someone please confirm this for me. In category theory one defines monoidal categories. One can then consider monoids in a monoidal category $C$. Given such a monoid $M$, one can then consider ...
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Monoidal categories: strictness from coherence

A version of Mac Lane's Coherence Theorem states that every formal diagram (i.e., a diagram that involves only the associativity isomorphism, the unit isomorphisms, their inverses, identity morphisms, ...
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What is the isomorphism from the identity functor to the double dual in a symmetric monoidal $(\infty,1)$-category?

I'm following example 2.4.12 in Lurie's note on the classification of TFTs. Let $\mathcal{C}$ be a symmetric monoidal $(\infty,1)$-category that has duals. The action of $O(1)$ on the largest $\infty$-...
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The Eckmann-Hilton argument gives us an isomorphism between $Mon(Mon(C))$ and $CoMon(C)$ or just an equivalence?

I'm studying symmetric monoidal categories and I have seen some authors saying that, due to the Eckmann-Hilton argument, given some symmetric category $C$, the category $Mon(Mon(C))$ of monoidal ...
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Completeness of the category of enriched categories

In the nLab entry on strict n-categories, one reads: For $V$ any complete and cocomplete closed monoidal category, also $VCat$ (the category of V-enriched categories) has these same properties. Is ...
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Monoidal categories as categories internal to $\textbf{Cat}$

For any category $C$, there is a terminal functor $C\to \ast$ to the singleton category. With this, a monoidal category $(C,\otimes,I)$ has natural candidates to make it a category internal to $\...
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Strict 2-groups and group objects in $\textbf{Cat}$

Group objects in $\textbf{Cat}$ are strict monoidal categories with an antipode functor endofunctor $\text{inv}$ such that the standard diagram of groups (with the appropriate replacements), shown ...
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Linear algebra in Rel

I'm interested in generalising some of the basics of linear algebra. In a standard setup, we take an arbitrary semiring $R$, and from this can make vectors as elements of $R^n$ and matrices as ...
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If the monoidal categories of $R$-bimodules and $S$-bimodules are monoidally equivalent, must $R$ and $S$ be Morita equivalent?

If $R$ and $S$ are Morita equivalent rings, then the monoidal categories of $R$-bimodules and $S$-bimodules are monoidally equivalent. Now, consider the converse: Question: If the monoidal ...
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Definition of tensor product of graded maps

The tensor product of $\mathbb{Z}$-graded vector spaces $V = \bigoplus_iV_i$ and $W = \bigoplus_iW_i$ is $$ V\otimes W = \bigoplus_i(V\otimes W)_i, $$ where $$ (V\otimes W)_i=\bigoplus_{j\leq i}(V_j\...
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Motivation/Intuition for the Pentagon Axiom

I have just started reading a bit on monodical categories, and there is I just can't make much sense of: the Pentagon Axiom. To provide some context, we have a category $\mathcal{C}$ together with a ...
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Constructing a monoidal structure in a closed bicategory

I wanted to know whether there was a bicategorical version of Eilenberg-Kelly's theorem that allows to reconstruct a monoidal structure on a closed category. Explicitely, if a category $C$ is closed ...
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Are $(\mathscr{C}\times \mathscr{C})\times \mathscr{C}$ and $\mathscr{C} \times (\mathscr{C} \times \mathscr{C})$ considered equal?

I'm familiar with monoidal categories and want to make sure I completely understand the definition. One thing I keep getting stuck on is the natural isomorphism between $(X\otimes Y)\otimes Z$ and $X\...
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What is the name of this identity relating the monoidal products of finite sets and finite-dimensional vector spaces?

Note: This question is almost certainly a duplicate. Since I don't know the terminology involved, I couldn't find the original question. If someone can find the original question and link to it, then ...
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Hopf monoids in different categories

Hopf algebras are precisely Hopf monoids in the category of vector spaces. What are Hopf monoids in other common (to be interpreted by the reader) monoidal categories? In particular, are Hopf ...
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Does this “leaky isomorphism” concept have a name?

I've been thinking about catalysis in chemical reaction networks while learning category theory at the same time, and it's given me a weird idea, which I'm asking about out of curiosity. Suppose I ...
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(Co)modules in arbitrary monoidal categories

To make sure we are using the same definitions: In any monoidal category (C, $\otimes$, I) we have a notion of a monoid object (M, $\mu$, $\eta$). This is an object M $\in$ C with morphisms $\mu$: M $...
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The category of monoid objects

Let $C$ be a cartesian category. We can form the category $\text{Mon}(C)$ of monoid objects in $C$. If $C$ is $\text{Set}$, this category is the category of monoids. If $C$ is $\text{Mon}$ (the ...
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Endofunctors as expression builders

I was recently reading through Bartoz Milewski's blog posts on category theory on monads and the Eilenberg-Moore algebra(see here). He mentions that if $T:C \to C$ is an endofunctor then it can be ...
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tensor product and composition compatibility in a monoidal category

Let $C$ be a strict monoidal category. Take objects, $a,b\in C$ and morphisms $T\in C(a,1)$, $S\in C(1,b)$. Does it hold then that $ST=S\otimes T=T\otimes S$? In particular, does the identity morphism ...
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Rigid structure for the category of infinite-dimensional vector spaces

The category of finite-dimensional vector spaces, endowed with its usual tensor product, is rigid, that is it admits right and left duals. What happens in the infinite-dimensional case?
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Categorification of $Vec_G$

Let $G$ be a finite abelian group. We can think of the monoidal category $1Vec_G=Vec_G$ of $G$-graded $k$-vector spaces as a categorification of the group algebra $0Vec_G=kG$. How does this ...
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Additive Monoidal functor between categories of graded vector spaces

While reading the Tensor categories book i realised that there were mistakes in section 2.6. Thiel in this https://ulthiel.com/math/teaching-org/tensor-categories/#Chapter_2_Monoidal_categories ...
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functor between the category of Finite sets which is not monoidal

If I consider the category of finite sets with the monoidal product defined by the cartesian product are there endofunctors which are not monoidal functors? if so I was wondering if there is a ...
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Rigid structure of $\operatorname{Vec}(G)$

Apparently if $G$ is a finite group $\operatorname{Vec}(G)$ is a Rigid monoidal category (whereby I mean the category of graded vector spaces over a group $G$). I struggle to see why it is rigid. It ...
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Monoidal categories: coherence from strictness

In some modern textbooks (e.g., Tensor Categories by Etingof et al. and Brandenburg's textbook on category theory which is in German), Mac Lane's coherence theorem is deduced from Mac Lane's ...
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If two monoidal categories are monoidally isomorphic then if one is strict so is the other.

I am trying to show this fact as it seems to be assumed in Remark 2.8.6 of the tensor categories book. Any help would be appreciated. Recall that a category is strict if the associator is the ...
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Tensor product of bialgebras in a braided monoidal category

Let $(\mathcal{C},\otimes,\mathbb{I},\alpha,\lambda,\rho,c)$ be a braided monoidal category. By this I mean a (not necessarily strict) monoidal category $(\mathcal{C},\otimes,\mathbb{I},\alpha,\lambda,...
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Monoidal functor and the units II

In their book Tensor Categories Etingof, Gelaki, Nikshych and Ostrik give a different definition of a (strong) monoidal functor. The difference is that they do not set the isomorphism $F(1) \cong 1$ ...
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Monoidal functor and the units

In their book Tensor Categories Etingof, Gelaki, Nikshych and Ostrik define a monoidal functor between monoidal categories $(C,\otimes,1,\alpha,r,s)$ and $(C',\otimes', 1',\alpha',r',s')$ as the ...
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Strong monoidal equivalence from a symmetric monoidal category to a strict monoidal category

This question is related to point $(1)$ in this answer: https://math.stackexchange.com/a/190402/229776 Let $(\mathsf{C}, \otimes, 1, \alpha, l ,r, s)$ be a symmetric monoidal category, $(\mathsf{C}...
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Coherence for monoidal categories - the proof of Mac Lane

I will use the notation from Mac Lane's Categories for the Working Mathematician. To avoid confusion and possible mistakes, I will add screenshots of the material in question, rather than trying to ...
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Is the bifunctor of a monoidal category faithful?

Let $\otimes: C\times C\to C$ be the bifunctor of a monoidal functor $C$ with left unitor $\rho$ and right unitor $\lambda$, and identity object $e$. By using the naturality of $\rho$ I can show that ...
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How do the coherence conditions for a monoidal category imply “associativity of the monoidal product”

Intuitively, I would say that "associativity of the monoidal product" should mean: for all objects $A,B,C$, there is a natural isomorphism so that $(A\ast B)\ast C \cong A\ast (B\ast C)$, and for all ...

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