Skip to main content

Questions tagged [monoidal-categories]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
45 views

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 ...
Buschi Sergio's user avatar
6 votes
0 answers
114 views

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 $\infty$-...
John Nolan's user avatar
6 votes
0 answers
105 views

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], ...
Minkowski's user avatar
  • 581
3 votes
0 answers
66 views

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 ...
Léo S.'s user avatar
  • 193
0 votes
0 answers
73 views

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,...
shamwowexcitante's user avatar
16 votes
0 answers
594 views

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 ...
Dmitri Pavlov's user avatar
6 votes
0 answers
97 views

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 \...
varkor's user avatar
  • 9,531
6 votes
0 answers
116 views

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 ...
Tim Campion's user avatar
  • 62.7k
6 votes
0 answers
90 views

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 ...
jonb25's user avatar
  • 61
1 vote
0 answers
26 views

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 ...
AT0's user avatar
  • 1,447
4 votes
1 answer
168 views

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 ...
Simon Henry's user avatar
  • 40.8k
5 votes
3 answers
398 views

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 ...
May's user avatar
  • 110
1 vote
1 answer
158 views

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 $\...
Sebastien Palcoux's user avatar
6 votes
1 answer
354 views

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 ...
Bobby-John Wilson's user avatar
2 votes
1 answer
153 views

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': ...
Yilmaz Caddesi's user avatar
3 votes
2 answers
198 views

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 ...
Yilmaz Caddesi's user avatar
3 votes
1 answer
189 views

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....
user39598's user avatar
  • 499
3 votes
0 answers
38 views

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 ...
Léo S.'s user avatar
  • 193
5 votes
3 answers
399 views

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{...
Xiong Jiangnan's user avatar
3 votes
0 answers
96 views

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}) = \...
NoetherNerd's user avatar
3 votes
2 answers
114 views

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(...
Lagrenge's user avatar
  • 575
5 votes
1 answer
169 views

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 ...
Emily's user avatar
  • 11.5k
3 votes
1 answer
117 views

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 ...
YjL's user avatar
  • 41
2 votes
1 answer
190 views

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 ...
SetR's user avatar
  • 91
3 votes
0 answers
58 views

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 $\...
Nikio's user avatar
  • 351
4 votes
1 answer
129 views

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 ...
willie's user avatar
  • 499
3 votes
0 answers
91 views

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 $(V,\rho,\Delta)$ ...
Max Demirdilek's user avatar
5 votes
1 answer
145 views

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 ...
Milo's user avatar
  • 53
1 vote
1 answer
133 views

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 ${}^...
Yilmaz Caddesi's user avatar
5 votes
1 answer
371 views

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 ...
Anthony D'Arienzo's user avatar
3 votes
1 answer
188 views

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 ...
Yilmaz Caddesi's user avatar
4 votes
0 answers
140 views

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 $[-,-]$ ...
Max Demirdilek's user avatar
3 votes
0 answers
121 views

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 ...
Max Demirdilek's user avatar
4 votes
1 answer
178 views

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?
Cameron Zwarich's user avatar
6 votes
0 answers
172 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
7 votes
0 answers
269 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
  • 15.5k
3 votes
0 answers
199 views

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 ...
E. KOW's user avatar
  • 752
7 votes
0 answers
232 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
3 votes
0 answers
86 views

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 $X\in \mathcal{V}$ ...
Max Demirdilek's user avatar
5 votes
1 answer
368 views

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,...
Max Demirdilek's user avatar
5 votes
1 answer
259 views

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 ...
varkor's user avatar
  • 9,531
2 votes
1 answer
142 views

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 \...
Yilmaz Caddesi's user avatar
3 votes
1 answer
268 views

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 ...
Yilmaz Caddesi's user avatar
4 votes
1 answer
349 views

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 ...
Martin Brandenburg's user avatar
1 vote
1 answer
280 views

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 ...
Andrew's user avatar
  • 121
2 votes
1 answer
214 views

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 \...
Didier de Montblazon's user avatar
-1 votes
1 answer
170 views

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 ...
seldon's user avatar
  • 1,053
8 votes
1 answer
248 views

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 ...
Lagrenge's user avatar
  • 575
3 votes
0 answers
112 views

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 ...
xuq01's user avatar
  • 1,056
7 votes
0 answers
290 views

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

1
2 3 4 5
11