Questions tagged [monoidal-categories]

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Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following: The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
Gerrit Begher's user avatar
19 votes
0 answers
445 views

monoidal (∞,1)-categories from weakly monoidal model categories

In Higher Algebra section 4.1.7, Jacob Lurie proves that the underlying $(\infty,1)$-category of a monoidal model category is a monoidal $(\infty,1)$-category. Dominic Verity and Yuki Maehara have (...
Emily Riehl's user avatar
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16 votes
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339 views

Specific cases of the tangle hypothesis in terms of "classical" n-categories

As is well known, the tangle hypothesis of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal $n$-...
Dominic Else's user avatar
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1k views

Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether the category of commutative or noncommutative C*-algebras or von Neumann algebras is equivalent to the category of commutative or ...
Dmitri Pavlov's user avatar
16 votes
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2k views

What is known about module categories over general monoidal categories?

All of the literature I have seen on module categories over monoidal categories has been in the rigid $k$-linear semisimple case, more or less in the spirit of Ostrik's paper, Ostrik, V. Module ...
Evan Jenkins's user avatar
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12 votes
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410 views

Biased vs unbiased lax monoidal categories

There are two principal ways to define a monoidal category: The biased definition includes a unit object $I$, a binary tensor product $A\otimes B$, and a ternary associativity isomorphism $(A\otimes ...
Mike Shulman's user avatar
10 votes
0 answers
117 views

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 ...
varkor's user avatar
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Subobject classifiers with a quantale structure

Quantales were kind of popular a few decades ago, as some sort of "quantum" version of the much more widespread concept of locale (Mulvey introduced them to study $C^*$ algebras). I was ...
Mirco A. Mannucci's user avatar
10 votes
0 answers
372 views

Internal logic in topos theory, monoidal categories, and quantum mechanics

To obtain the internal logic of a topos (roughly speaking), we associate a type of free variable with an object, and a statement about such a variable with a subobject of that object. Intuitively, the ...
Neuromath's user avatar
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What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?

A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...
მამუკა ჯიბლაძე's user avatar
9 votes
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262 views

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 ...
Antoine Labelle's user avatar
9 votes
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147 views

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)....
Tim Campion's user avatar
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Explicit description of the tensor product of symmetric monoidal categories / Picard groupoids

$\newcommand{\lax}{\mathsf{lax}}\newcommand{\oplax}{\mathsf{oplax}}\newcommand{\str}{\mathsf{str}}\renewcommand{\S}{\mathbb{S}}\newcommand{\F}{\mathbb{F}}\newcommand{\Hom}{\mathrm{Hom}}$We have tensor ...
Emily's user avatar
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9 votes
0 answers
254 views

t-structures on the tensor product of stable $\infty$-categories

It is a matter of checking definitions that the tensor product of presentable $\infty$-categories restrict to a tensor product between stable presentable $\infty$-categories; I was wondering if there ...
fosco's user avatar
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Twisted duality in a symmetric monoidal category

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples? Definition. Let $\mathcal{C}$ be a ...
Martin Brandenburg's user avatar
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218 views

Are pivotal categories the algebras for a cartesian monad?

It seems to be "known" but not written down that the following are more-or-less equivalent: ...
Bruce Westbury's user avatar
8 votes
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100 views

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 $C$ ...
Tim Campion's user avatar
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8 votes
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156 views

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 ...
Branimir Ćaćić's user avatar
8 votes
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299 views

How to define the direct sum of TQFTs $(\infty,1)$-categorically?

Let $\mathit{Bord}_d$ be the symmetric monoidal category of $(d-1)$-manifolds and bordisms between them. Let $\mathcal{C}$ be the symmetric monoidal category of $k$-modules. Then, for two symmetric ...
Jan Steinebrunner's user avatar
8 votes
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211 views

Categorical interpretation of quantum double $D(A,B,\eta)$

It is known that the Drinfel'd double $D(A)$ of a Hopf algebra $A$ is characterized by the following two properties: The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the ...
Bipolar Minds's user avatar
8 votes
0 answers
269 views

Practice of higher categories - giving rigorous constructions

1) Let $\mathcal{C}$ be a monoidal $\infty$-category, $A$ an algebra object in $\mathcal{C}$, and $M$ a left $A$-module in $\mathcal{C}$. So, in Lurie's formalism, $\mathcal{C}$ is encoded by some ...
Sasha's user avatar
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Which Drinfeld centers are balanced monoidal, i.e. have a twist?

A twist is an automorphism $\theta$ of the identity functor of a monoidal category with braiding $c$, such that $\theta_{X \otimes Y} = c_{Y,X} c_{X,Y} (\theta_X \otimes \theta_Y)$. A braided monoidal ...
Manuel Bärenz's user avatar
7 votes
0 answers
248 views

What exactly is a Tannakian subcategory?

I've searched all the standard references (Deligne--Milne, Saavedra-Rivano) and cannot find a definition of Tannakian subcategory. What I find is many authors who discuss the Tannakian subcategory ...
David Corwin's user avatar
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Does the pentagon axiom force the associativity constraint to be a natural isomorphism?

Consider a fusion ring and the associated system of polynomial equations induced by the pentagon axiom of a fusion category. A solution of this system is supposed to encode the associativity ...
Sebastien Palcoux's user avatar
7 votes
0 answers
165 views

Who first introduced the term "categorical group", and when?

The term "categorical group" is often used to mean a group object in Cat; these days we also call such a thing a strict 2-group. Who first introduced the term "categorical group", ...
John Baez's user avatar
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7 votes
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297 views

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 ...
Bruno Gavranovic's user avatar
7 votes
0 answers
165 views

Strictifying monoidal 2-functors

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a (weak) monoidal 2-functor between two strict monoidal 2-categories. Up to replacing $\mathcal{C}$ by an equivalent strict monoidal 2-category, can I ...
JeCl's user avatar
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370 views

Left Kan extensions of "strong" monoidal functors

Consider the 2-category $\mathsf{MonCat}$ where objects are monoidal categories, 1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations. Given arrows $f: \mathsf{C} \to ...
Eigil Fjeldgren Rischel's user avatar
7 votes
0 answers
141 views

Strictifying closed monoidal categories?

Let $C$ be a cartesian closed category. It's well known that $C$ is equivalent to a category where the product is strict monoidal; i.e. where there are equalities of the functors given by the ...
ClosedCoherence's user avatar
7 votes
0 answers
317 views

Non-linear Galois descent

This question is about Galois theory. So let $K / k$ be a Galois extension of fields. Let us assume that $K / k$ is finite dimensional, though everything can be made to work in the profinite case by ...
Jakob Werner's user avatar
  • 1,093
7 votes
0 answers
283 views

Simple groups up to outer automorphisms

The following wonderful paper: Manfred Droste, Michèle Giraudet, Rüdiger Göbel, All Groups are Outer Automorphism Groups of Simple Groups Journal of the London Mathematical Society 64 (2001) 565–...
Bedovlat's user avatar
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7 votes
0 answers
170 views

When is Rep(U_q(g)) invariant under q -> -q and why?

Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...
Noah Snyder's user avatar
  • 27.8k
6 votes
0 answers
150 views

Drinfeld center of non-rigid closed monoidal categories

Context. The Drinfeld center of a rigid monoidal category $\mathcal{C}$ is again rigid. This is not hard to see: Given an object $X\in \mathcal{C}$ together with a half-braiding $\phi_X:X\otimes-\...
Max Demirdilek's user avatar
6 votes
0 answers
184 views

Does a fusion ring with F and R-symbols uniquely determine a braided tensor category?

Background : In mathematical physics, 'anyons' in (2+1) dimensional systems are described by braided tensor categories. The anyon types correspond to the irreducible objects of the category. From such ...
Alex_Bols's user avatar
6 votes
0 answers
845 views

Tannaka without Yoneda?

I am studying enriched categories, and as I wrote in my previous question How is the morphism of composition in the enriched category of modules constructed?, this is very difficult because there are ...
Sergei Akbarov's user avatar
6 votes
0 answers
145 views

$\mathbb{E}_\infty$-refinements of the graded tensor product of $\mathbb{Z}$-graded spectra

The category $$\mathsf{Gr}_\mathbb{Z}\mathsf{Mod}_R\overset{\mathrm{def}}{=}\mathsf{Fun}^\otimes(\mathbb{Z}_\mathsf{disc},\mathsf{Mod}_R)$$ of $\mathbb{Z}$-graded $R$-modules has a natural monoidal ...
Emily's user avatar
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6 votes
0 answers
154 views

Has this notion of a ring in a bimonoidal category been studied before?

The Baez–Dolan microcosm principle is stated in the nLab as follows. Microcosm principle: Certain algebraic structures can be defined in any category equipped with a categorified version of the same ...
Emily's user avatar
  • 10.3k
6 votes
0 answers
463 views

"Fundamental theorem for Hopf modules"

I am studying Hopf algebras in categories, and I hope, somebody could help me with the following. Joost Vercruysse in his paper Hopf algebras---Variant notions and reconstruction theorems writes (...
Sergei Akbarov's user avatar
6 votes
0 answers
216 views

Can C*/W*-algebras be realized as (involutive?) monoid/co-monoid objects?

I would like to know how close one can get to realizing the category of C*-algebras as a category of monoid objects. Related (almost, but not quite, duplicate) questions are: "Recovering a monoidal ...
Tom Mainiero's user avatar
6 votes
0 answers
166 views

Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...
Will Sawin's user avatar
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6 votes
0 answers
338 views

Compatibility between strength and costrength of a monoidal monad

Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A ...
Martin Brandenburg's user avatar
6 votes
0 answers
719 views

Tensor product of quivers

As a special case of a general construction I have constructed "accidentally" a tensor product of quivers aka directed multigraphs (aka directed graphs for category theorists). Probably this ...
Martin Brandenburg's user avatar
6 votes
0 answers
343 views

Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.

In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation: Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every ...
Gerrit Begher's user avatar
5 votes
0 answers
205 views

Lax monoidal structure on the right Kan extension of a partially monoidal Γ-set

First some preliminaries. Let me write $Fin_\ast$ for the skeleton of the category of finite pointed sets and pointed maps between them on the objects $n_+=\{0,1,...,n\}$, where $0$ is the base point (...
Jonathan Beardsley's user avatar
5 votes
0 answers
77 views

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 $\...
Kevin Walker's user avatar
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5 votes
0 answers
93 views

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 ...
Thibault Décoppet's user avatar
5 votes
0 answers
251 views

Strictification for closed monoidal categories

The strictification theorem for monoidal categories states that every monoidal categorically is monoidally equivalent to a strict monoidal category. Is there a strictification theorem for closed ...
varkor's user avatar
  • 8,675
5 votes
0 answers
116 views

Variations on Thomason's equivalence between connective spectra and symmetric monoidal categories

There's a number of results relating monoidal categories to connective spectra (which are themselves equivalent to $\mathbb{E}_{\infty}$-spaces): Symmetric monoidal categories model all connective ...
Emily's user avatar
  • 10.3k
5 votes
0 answers
150 views

Morita equivalence for monoidal categories

I came across this MO post and it got me thinking. In monoidal categories we can define module and bi-module categories, so what can we say about morita equivalence in this situation? Which of the ...
misseuler's user avatar
5 votes
0 answers
222 views

Derived category of an abelian monoidal category

For any abelian category $\mathcal{A}$, we can consider its derived category $\mathcal{D(A)}$, which is naturally triangulated. If $\mathcal{A}$ is endowed with a monoidal structure (bilinear with ...
Dick Johnson's user avatar