# Questions tagged [monoidal-categories]

The monoidal-categories tag has no usage guidance.

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### Tamari lattice and bicategory coherence

Reference links:
Tamari lattice (Wikipedia): https://en.wikipedia.org/wiki/Tamari_lattice
Associahedra: https://en.wikipedia.org/wiki/Tamari_lattice#/media/File:Tamari_lattice.svg
The Tamari lattice ...

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### Is there a synthetic approach to (symmetric) monoidal infinity-categories?

Recent work of Riehl and Verity (e.g. the book "Elements of $\infty$-category theory") has established a "synthetic" / model-independent approach to the study of $\infty$-...

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### Equivalence between bialgebras and finite ring categories with fibre functor

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Rec{Rec}\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\newcommand\fdMod[1]{#1\text-{\Mod}^\text{fd}}$In the celebrated [EGNO, Thm 5.2.3], ...

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### Enriched tensor product of chain complexes

Question (idea): Is there a notion of tensor product of chain complexes in a $\mathcal{V}$-enriched monoidal category $\mathcal{C}$, for $\mathcal{V}$ a linear symmetric monoidal category?
Let me ...

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### Coherence between associator and unitor in a monoidal category

I have a hard time trying to prove a coherence relation in a monoidal category. I'll start by giving the definition I'm working with:
A monoidal category is a sextuple $(C, \otimes, \mathbf{1}, \alpha,...

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### Are dualizable topological vector spaces finite-dimensional?

Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...

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### Is the symmetry compatibility condition in Fox's theorem necessary?

Let $(\mathscr V, \otimes, 1, \sigma)$ be a symmetric strict monoidal category whose unit is terminal. Suppose that every object $A$ is equipped with the structure of a cocommutative comonoid $1 \...

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### If a strong monoidal functor $F$ has an ambidextrous adjoint, then how close is the adjoint to being strong monoidal?

Let $F : C \to D$ be a strong (symmetric, say) monoidal functor. Suppose that $G : D \to C$ is both left and right adjoint to $F$ (an ambidextrous adjunction). Then by doctrinal adjunction $G$ is both ...

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### Permutative Gray monoid that can not be strictified

Strict monoidal 2-categories do not suffice to model all connected 3-types: the non-trivial coherence cells $\Sigma: (f \otimes 1)\circ (1 \otimes g) \cong (1 \otimes g)\circ (f \otimes 1) $ of a Gray ...

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### Equivalences induced from invertible objects in transported bifunctors along an adjoint pair

I'm interested in the following problem, similar in vein to this other question. To put it simply, I have an adjoint pair $F\dashv G$ between categories $\mathrm{C}$ and $\mathrm{D}$ and I suppose ...

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### How general is $TX \otimes X \simeq X \otimes TX$?

Let $C$ be a monoidal category, $X \in C$ an object, and $TX$ the free monoid object on $X$, assuming it exists. How often do we have an isomorphism $TX \otimes X = X \otimes TX$? is there a canonical ...

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### Group completion of a monoid (braid groups)

Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups.
For a discrete group $G$, we let $BG$ to be the classifying space of $G$.
After reading this question, I was ...

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### Are the minimal nondegenerate extensions universal?

We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...

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### Endomorphisms of simple dualizable objects in a linear abelian monoidal categories

In When is an object in a linear or abelian category simple? Or: How should I define fusion categories? endomorphisms of simple objects in a k-linear abelian category are discussed. In the answer it ...

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### In a monoidal category with duals is the coevaluation map determined by the evaluation?

For a monoidal category $(\mathcal{M},\otimes,1)$ and an object $X$ with a left dual $X^*$. Let $(e,c)$ be a pair of (co)evaluation maps for the pair $(X,X^*)$. Is it possible to have another map $c': ...

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### The evaluation and coevaluation maps for an object isomorphic to a dualisable object

Let $X$ be an object in a monoidal category with dual $X^*$ and evaluation and coevaluation maps $e$ and $c$. Now if we have an isomorphism $\sigma:X \to Y$, for some other object $Y$, then $Y$ must ...

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### Monoidal structure on presheaves

I am confused about the following monoidal structure, which gives a symmetric monoidal structure on R-modules (that I think is not Cartesian), even if R is not commutative.
Let $C$ be a small category....

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### Coherence of the graphical language for pivotal categories

Throughout I follow A survey of graphical languages for monoidal categories, Peter Selinger, arXiv.
A pivotal category is a monoidal category where each object $A$ has a dual $A^*$, together with a ...

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### Bar construction in commutative algebras is calculated by pushout

$\DeclareMathOperator\colim{colim}$
Also asked in MathStackexchange here
This is a statement in Lurie's Higher Algebra 5.2.2.4.
Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...

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### Isomorphic objects have the same dimension (pivotal categories)

I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e.,
$$
\mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \...

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### Are the Drinfeld doubles of twist equivalent Hopf algebras twist equivalent?

Let $H_1$ and $H_2$ be finite dimensional Hopf algebras that are twist equivalent, i.e. $H_2$ is obtained from $H_1$ using a Drinfeld twist. My question is: are the Drinfeld doubles $D(H_1)$ and $D(...

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### 1-categorical universal properties for the smash product of pointed sets

Question I. Is the following statement, inspired by this one, true?
Universal Property I. The symmetric monoidal structure on the category $\mathsf{Sets}_*$ of pointed sets and pointed maps between ...

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### Different notions of equivalences of $\mathcal{O}$-monoidal $\infty$-categories

I am currently reading Higher Algebra by Jacob Lurie and I have a question regarding equivalences of $\mathcal{O}$-monoidal categories.
Let $\mathcal{O}$ be an $\infty$-operad. Suppose that I have two ...

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### Is the category of simplicial $R$-modules closed monoidal?

I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state ...

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### Self-enrichment for a closed monoidal bicategory

First, there are two possible generalization of the notion of closed category, vertical and horizontal.
I'm interested in the vertical one, something saying, I guess, that a monoidal bicategory $\...

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### Does the Gray tensor product exhibit Gray as a monoidal Gray-category?

Crans constructs a lax Gray tensor product for strict omega-categories that defines a monoidal structure on the category of strict omega-categories $\omega Cat$. We write Gray for this monoidal ...

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### Yetter-Drinfeld modules for Hopf monads

1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ ...

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### One-object lax natural transformation

A lax natural transformation $\alpha$ between two pseudofunctors $F,G: \mathcal{K} \to \mathcal{L}$ between bicategories $\mathcal{K}, \mathcal{L}$ consists of the following data:
For every object $A ...

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### Dual objects in an abelian monoidal category

Let $(\mathcal{M},\otimes)$ be a monoidal category that is also abelian, and assume that $\otimes$ is bilinear with respect to the biproduct. Let $X$ and $Y$ be two objects that admit (left) duals ${}^...

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### Day convolution and sheafification

$\DeclareMathOperator\Psh{Psh}\DeclareMathOperator\Sh{Sh}\newcommand\copower{\mathrm{copower}}$I was looking through Bodil Biering's thesis On the Logic of Bunched Implications - and its relation to ...

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### Left duals and right duals are also isomorphic in a semisimple category

In the n-Lab page
https://ncatlab.org/nlab/show/rigid+monoidal+category
it is written that
Left duals and right duals are also isomorphic in a semisimple
category.
For a left dual semisimplicity ...

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### Examples of $\ast$-autonomous $(\infty,1)$-categories

A $\ast$-autonomous category is a biclosed monoidal category together with a dualizing object. An object $\bot$ in a biclosed monoidal category $(\mathcal{C},\otimes)$ with left internal hom $[-,-]$ ...

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### Tannaka duality for Hopf algebroids

Setting. Let $k$ be a field, $A$ a finite-dimensional $k$-algebra, and $H$ a Hopf algebroid over $A$ with invertible antipode. Denote by $\operatorname{mod}(H)$ the category of finite-dimensional ...

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### Monoidal topology and coarse spaces

Is there a description of (quasi-)coarse spaces that is analogous to the description of (quasi-)uniform spaces as lax algebras?

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### 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-\...

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### 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 ...

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### Coevaluation for linear categories

For a field $k$ and an associative $k$-algebra $R$, the $k$-linear category $R\operatorname{-Mod}$ is self dual inside $\operatorname{DGCat}_k$, with the counit map sending $k$ to $R$ regarded as a ...

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### 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 ...

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### Tensor product of functors, central Hopf monad and star-autonomy

Setting.
Let $\mathcal{C}$ be a category and $(\mathcal{V},\otimes,I, \multimap)$ a (symmetric) closed monoidal category. Let $F:\mathcal{C}\rightarrow \mathcal{V}$ be a functor and $X\in \mathcal{V}$ ...

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### A result on symmetric closed monoidal categories

In this discussion from the categories mailing there is mention of the following result by Robin Houston, supposedly proved in 2006:
Theorem. Let $\mathcal{C}$ be a symmetric closed monoidal category,...

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### 3-functoriality of the lax Gray tensor product

In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For ...

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### Duality in a monoidal category as a functor

In a rigid monoidal category $\mathcal{M}$ every object has a (say left) dual. Is the process of taking duals functorial? More specifically - is there a well-defined functor
$$
\mathcal{M} \to \...

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### A question about rigid objects in monoidal categories

Let $(\mathcal{A},\otimes,1_{\mathcal{A}})$ be a monoidal category, and let $M$ be a rigid object of $M$, with left and right dual respectively denoted by
$$
\Big\{M^*,~~~ \epsilon_l:M^* \otimes M \to ...

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### Does the category of integral domains admit a symmetric monoidal structure?

Let $\mathbf{Int}$ be the category of integral domains with injective homomorphisms. Does it admit a symmetric monoidal structure? If so, can we choose $\mathbb{Z}$ as the unit object? If it helps to ...

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### What are the morphisms in the category of retractions?

In Michael Shulman's Framed bicategories and monoidal fibrations Example 12.10 he defines a category $\operatorname{Retr}(\mathcal{C})$ as the "category of retractions in $\mathcal{C}$". He ...

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### Isomorphisms after tensoring with the identity in a monoidal category

Let us take the following assumptions: $\mathscr{M}$ a monoidal category, $X,Y,Z$ three objects in the category, and $f: Y \to Z$ a morphism. If the morphism
$$
\mathrm{id}_X \otimes f: X \otimes Y \...

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### Does a symmetric monoidal functor between cartesian monoidal categories automatically preserve products? [closed]

Suppose $\cal A, B$ are cartesian monoidal categories, that is, categories equipped with a choice of finite products (including the nullary one, $1$). Suppose, moreover, that $F: \cal A \to B$ is a ...

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### How does the Tannaka duality work for weak Hopf algebras and fusion categories?

I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works ...

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### Braided monoidal categories as generalized "braided" schemes

It's well know by the Gabriel-Rosenberg reconstruction theorem that a (quasi-separated) scheme $X$ is completely determined by its category of quasicoherent sheaves $\mathbf{QCoh}(X)$. The latter is ...

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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 ...