Questions tagged [monoidal-categories]
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492
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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, ...
- 510
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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 ...
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