# Questions tagged [monoidal-categories]

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### Why is $\rm{Cat}$ a Cartesian-closed category?

I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories.
Two general examples:
Grothendieck topos with Cartesian structure. Here, for example, $\...

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### Is there a notion of "knot category"?

Consider a rigid braided monoidal category, with braiding $\beta_{x,y} : x \otimes y \cong y \otimes x$, and every object has a dual such that $\epsilon_x : 1 \to a \otimes a^*, \bar\epsilon_x : a^* \...

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### Abelian categories that are not monoidal

Forgive me if this turns out to be a naive question. I'm quite convinced that not all abelian categories admit (symmetric?) monoidal structure (of course, with the tensor product being additive, ...

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### Relationship between fusion category and its Drinfel'd center

Is it true that given a fusion category $\mathcal{C}$ and its Drinfel'd center $Z(\mathcal{C})$, there is a fully faithful functor $F:\mathcal{C}\hookrightarrow Z(\mathcal{C})$? I.e. can $\mathcal{C}$ ...

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### "Closed $\mathscr{V}$-modules are uniquely (co)tensored $\mathscr{V}$-categories": shouldn't we assume they are also "mixed monoidal"?

$\newcommand{\M}{\mathcal{M}}\newcommand{\V}{\mathscr{V}}\newcommand{\hom}{\mathsf{hom}}$Throughout this post, $\V$ refers to some "cosmos", where I borrow the word "cosmos" from ...

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### Equivalence of Yetter-Drinfeld modules to Drinfeld center: is there a purely categorical proof?

Let $H$ be an Hopf algebra over a field $k$, and let $\mathcal{C}$ be the monoidal category of left $H$-modules.
It is known that the Drinfeld center of $\mathcal{C}$ is equivalent (as a braided ...

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### Proof that the unit of a Cartesian monoidal category is terminal

In short, given a monoidal category whose product is the categorical product, show that the unit object is terminal.
This looks very similar to questions that have been answered, but is subtly ...

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### Is there a generalization of braided monoidal category without the isomorphism requirement?

Is there a generalization of the notion of braided monoidal category that does not force the braiding $\gamma$ to be an isomorphism? I mean, it is of course possible to define such a kind of category, ...

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### Checking the triangle axiom of monoidal categories for the category $\operatorname{Vec}(\mathscr{C})$

Let $\mathscr{C}$ be a (for simplicity, strict) rigid $C^*$-tensor category. Consider the monoidal category $\operatorname{Vec}(\mathscr{C})$ as defined in section 2.4 of the article Operator algebras ...

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### Adding finite direct sums to a C*-tensor category

Consider the following fragment from the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset (p72, in section 2.5):
$\ \ \ $ Assume $\mathscr{C}$ is a ...

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### Algebra objects of $\operatorname{Vec}(\mathscr{C})$ are lax functors $\mathscr{C}^\text{op}\to \operatorname{Vec}$

Let $\mathscr{C}$ be a rigid $C^*$-tensor category. Let $\operatorname{Vec}(\mathscr{C})$ be the category with linear functors $\mathscr{C}^{\text{op}}\to \operatorname{Vec}$ (= category of complex ...

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### What is meant by saying that monoidal category of $U_h (sl_2 (\mathbb C))$ is different from that of $U(sl_2 (\mathbb C))[[h]]\ $?

I am following the notes on Quantum Groups given by our instructor. There I found that Drinfeld-Jimbo quantum group $U_h (sl_2 (\mathbb C))$ corresponding to the Lie algebra $sl_2 (\mathbb C)$ and the ...

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### Coproduct for a Frobenius algebra

The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...

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### Why is the category of strong braided functors from the braid category to a braided monoidal $M$ equivalent to the subcategory of *strict* functors?

This is my first, and probably my last, (for a while) posting on MO. I am very much a student and I don't claim to be a research mathematician, at all, but I have seen that sometimes "regular&...

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### A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?

A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...

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### Are the fusion categories weakly Frobenius?

A well-known open problem (generalizing Kaplansky 6th conjecture) asks whether every (spherical) fusion category $\mathcal{C}$ (over $\mathbb{C}$) is of Frobenius type, i.e. for every simple object $X$...

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### V-categories enriched in a monoidal V-category

In an email to the categories mailing list dated 21 August 2003, Street writes:
Max reminded me of his old result (not in the LaJolla Proceedings,
but known soon after) that a monoidal V-category is ...

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### $\ast$-autonomous categories with non-invertible dualizing object?

1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:
A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a ...

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### References for completions of finite group tensor categories

Let $G$ be a finite group and $\operatorname{Vec}_G$ be the tensor category of $G$-graded vector spaces (or, if you prefer, $\pi_{\le 2}(BG)$).
The completion $\overline{\operatorname{Vec}_G}$ of $\...

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### Transporting $\mathbb E_n$-monoidal structures between categories

Suppose given an $\mathbb E_n$-monoidal presentable $\infty$-category $\mathcal C$ (wrt. the Lurie tensor product $\otimes$), and $\mathcal D$ a presentable $\infty$-category. Suppose given a pair of ...

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### Are the morphisms of a star-autonomous category superfluous?

Let $(C,\otimes,I,\ast)$ be a (symmetric, say) star-autonomous category. Then $C$ comes equipped with a lax symmetric monoidal functor $|-|_C := Hom_C(I,-) : C \to Set$. The general hom-sets of $C$ ...

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### Full coherence for non-symmetric linearly distributive categories?

1. Context
Mac Lane's coherence theorem for monoidal categories can be phrased as "every formal diagram in a monoidal category commutes.“ I am interested in this type of coherence question for ...

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### Is this linearly distributive category really free?

In Natural deduction and coherence for weakly distributive categories Blute et al. claim to give a presentation of the free (non-symmetric) linearly distributive category $\operatorname{PNet_E}(C)$ on ...

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### Empires and the net criterion

Currently, I am struggling to understand the proof of Proposition 2.5 on page 250 (page 22 in the document) of the paper Natural deduction and coherence for weakly distributive categories by Blute, ...

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### Index-coclosure for monoidal categories, generalizing products and lextensive coproducts

I found a kind of monoidal structure that generalizes cartesian product and lextensive coproduct, and I'm wondering if anyone has seen it before and/or can tell me about it. I'm calling this structure ...

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### How does one classify monoidal biclosed structures on $Cat$?

Foltz, Kelly, and Lair assert that there are exactly two monoidal biclosed structures on the 1-category $Cat$ of small categories. But most of the proof is left as an "exercise" (see Prop 4)....

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### Separable monads do not induce separable monoids

Let us first recall the categorical notion of monad: if we have a category $\mathcal{C}$ then a monad on it consists in an endofunctor $\mathbb{A}\colon \mathcal{C}\rightarrow \mathcal{C}$ together ...

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### Right unitor in star-autonomous categories

1.Context
Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Here $r$ denotes the right unitor.
Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume ...

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### Cartesian monoidal star-autonomous categories

Disclaimer: This is a crosspost (see MathStackexchange). Apologies if cross-posting is frowned upon. However, it seems that on Stackexchange there are not many people familiar with star-autonomous ...

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### Star-autonomous categories are categorifications of Boolean algebras?

I asked this question fourteen days ago on MathStackexchange (see here). I have not received any answers or comments until now. It seems to me that on MathStackexchange not many people are familiar ...

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### Has anyone heard of a Lax Monoidal Functor where one of the arrows is flipped?

I've encountered the following scenario
I have a category $\mathcal C$ and for every object
$c\in\mathcal C$, I have found a monoidal category
$(\mathcal D, \otimes_c, 1_c,\cdots)$ such that I'm able ...

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### Strictification of $\mathcal{V}$-pseudofunctors

Let $\mathcal{B}$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $\textbf{st}\mathcal{B}$ and a ...

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### Isomorphism between Davydov-Yetter complex and Hochschild complex of canonical algebra on a multitensor category

I'm trying to follow the proof of proposition 7.22.7 from
Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, Tensor categories, Mathematical Surveys and Monographs 205. Providence, RI: ...

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### Constructing new categories by adding structure

On the one hand, let $\mathcal{C}$ be the monoidal category of finite-dimensional complex vector spaces and linear transformations, and on the other, let $\mathcal{D}$ be the monoidal category of ...

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### On the correspondence between proof nets and sequents

1. Context
While trying to answer my question on the existence of a (useful) graphical calculus for star-autonomous categories, I came across the paper Natural deduction and coherence for weakly ...

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### Transporting monoidal structure along adjunction

Let $L: \mathcal C \leftrightarrows \mathcal D : R$ be an adjoint pair and $\mathcal C = (\mathcal C,\otimes)$ be a monoidal category. I wonder under what conditions on the pair $L\dashv R$ we can ...

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### Left adjoints for functors out of a Deligne-Kelly tensor product

Let $k$ be a field, and let $\mathcal{C}$ be a locally finitely presentable $k$-linear categories. The Deligne-Kelly tensor product $\mathcal{C}\boxtimes\mathcal{C}$ is the 2-universal locally ...

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### Coherence theorem in braided monoidal categories

In MacLane's Categories for the working mathematician, the author shows that the evaluation at 1 gives an equivalence of categories $\mathrm{hom}_{\mathrm{BMC}}(B,M)\simeq M_0$ where $B$ is the braid ...

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### The coevaluation map for a projective module and its dual

$\DeclareMathOperator\coev{coev}$Let $R$ be a noncommutative ring and let $P$ be a bimodule over $R$, that is finitely generated and projective as a left module. It is "well-known" that any ...

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### Tensor categories with isomorphic Grothendieck semirings

Let $\mathcal{A}$ and $\mathcal{B}$ be two tensor categories whose Grothendieck semirings are isomorphic. Does it follow that the categories $\mathcal{A}$ and $\mathcal{B}$ are isomorphic (i) as ...

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### Does every monoidal category admit a braiding?

The question is in the title.
To make the statement more precise, is is true that for any given monoidal category $(\mathcal C, I, \otimes)$ there exists at least one braiding $\beta$? In other words, ...

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### Exactness of functors in a $C^*$-tensor category

I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. In section 2.3 "Fiber functors and reconstruction theorems*, the following ...

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### How many tensor products of chain complexes are there?

Let $Ch$ be the category of nonnegatively-(homologically-)graded chain complexes of abelian groups. Suppose that $(Ch,\boxtimes)$ is a monoidal biclosed structure.
Assume that the forgetful functor $(...

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### Enriched categories over a semi-monoidal category

Let $\mathcal{V}$ be a semi-monoidal category, meaning it satisfies the axioms of a monoidal category except missing a unit and the unit axiom. One could then still go about defining a $\mathcal{V}$-...

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### Pseudomonoids versus monoidal pseudofunctors from $\Delta$

I have been trying to find some literature (if there is any) on the relationship between pseudomonoids and monoidal pseudofunctors from the monoidal theory of a monoid, $\Delta$ (I am interested in ...

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### Cartesian product is to monoidal product as pullback is to what?

I'm trying to complete the following pattern
product : monoidal product : coproduct
pullback : ? : pushout
That is, if the monoidal product is a ...

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### The Kleisli category of a monoidal monad

Let $C$ be a symmetric monoidal category equipped with diagonals $\triangle_x: x \to x \otimes x$, that is, equipped with natural transformations $e_x: x \to 1$ and $\triangle_x : x \to x \otimes x $ ...

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### Non-semisimple categorification problem of fusion rings

We refer to [1] for the notions used in this post.
The Grothendieck ring of a fusion category (over $\mathbb{C}$) is a fusion ring, but there are fusion rings which are not of this form (...

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### "Partially strict" monoidal categories

Recall that a monoidal category $\newcommand{\C}{\mathcal{C}} (\C,\otimes,I)$ comes equipped with an associator $\alpha_{XYZ} \colon X \otimes (Y \otimes Z) \xrightarrow{\sim} (X \otimes Y) \otimes Z$ ...

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### Reference for free symmetric monoidal categories with duals on symmetric monoidal categories

The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons.
In ...