Questions tagged [monoidal-categories]

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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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8 votes
2 answers
233 views

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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9 votes
1 answer
551 views

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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2 answers
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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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2 answers
177 views

"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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This is not a tensor: tensoring abelian groups over groups

$\newcommand{\Cats}{\mathsf{Cats}}\newcommand{\MonCats}{\mathsf{MonCats}}\newcommand{\BrMonCats}{\mathsf{BrMonCats}}\newcommand{\SymMonCats}{\mathsf{SymMonCats}}\newcommand{\CMon}{\mathsf{CMon}}\...
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4 votes
1 answer
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Why is 'every braided monoidal category spacial'? [duplicate]

In his 2009 survey, Selinger ("A survey of graphical languages for monoidal categories") defines the notion of a 'spacial monoidal category', which (in his graphical calculus) allows one to ...
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2-morphism between circuits in a monoidal category

We are used to seeing equations between circuits in monoidal categories like this I am wondering about morphisms between string diagrams. I think they are 2-cells. I found an example of a 2-cell ...
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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 ...
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3 votes
1 answer
195 views

On the derived functor of the tensor product in a monoidal category

Let $(\mathcal{M},\otimes)$ be a symmetric monoidal model category; I'll assume for simplicity that every object is fibrant. Suppose that the unit $I$ is NOT cofibrant. I'm interested in whether the ...
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3 votes
1 answer
91 views

A non-projective rigid object in an abelian monoidal category

What is an example of a rigid object $A$ in an abelian monoidal category $\mathcal{M}$ that is not projective as an object in $\mathcal{M}$? (Since $\mathcal{M}$ is abelian projective just means that ...
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2 votes
0 answers
74 views

Existence of projective generators for a module category

Let $\mathcal{C}$ be a finite linear monoidal category and let $\mathcal{M}$ be a finite linear category which is a right module category over $\mathcal{C}$, i.e. there is a linear functor $ act : \...
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6 votes
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$\mathbb{E}_\infty$-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 ...
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5 votes
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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 ...
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5 votes
2 answers
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How is the morphism of composition in the enriched category of modules constructed?

I asked this a week ago at MSE, but without success. I am studying enriched categories and I have a feeling that I am doing something wrong because all the way each step, each elementary proposition, ...
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3 votes
1 answer
210 views

Corepresentability of involutory objects in monoidal $\infty$-categories

The group $\mathbb{Z}/2$ corepresents the functor $\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$ sending a monoid $A$ to its set of involutory elements (those satisfying $a^2=1_A$). A similar story ...
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1 vote
0 answers
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Delooping monoidal $(\infty,1)$-categories into $(\infty,2)$-categories

This is the one categorical level higher version of the question Delooping monoidal $\infty$-groupoids into $\infty$-categories. The classical, bicategorical, setting. Given a monoidal category $(\...
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2 votes
1 answer
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Delooping monoidal $\infty$-groupoids into $\infty$-categories

The classical setting. Given a monoid $A$, there's a category $\mathbf{B}A$, called the delooping of $A$, having a single object $\star$ and satisfying $\mathrm{Hom}_{\mathbf{B}A}(\star,\star)\overset{...
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2 votes
1 answer
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A twisted Haagerup category without pivotal structure

Let $G$ be a finite group, $\tau$ a group automorphism of $G$ of period two and $m$ a natural number. Following [1, Definition 2.1], a complex fusion category $\mathcal{C}$ is called a quadratic ...
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3 votes
1 answer
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Examples of semiring categories whose additive monoidal structure is not given by the coproduct

One of the most common examples of semiring categories is given by distributive monoidal categories. Indeed, examples of the latter include the following: $(\mathsf{CMon},\oplus,\otimes_{\mathbb{N}},...
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3 votes
0 answers
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What is the initial semiring category with a (commutative) semiring?

Recall that The biinitial monoidal category with a monoid is given by the augmented simplex category together with the monoid $([0],\sigma^{0}_{0},\delta^{0}_{0})$ there. The biinitial symmetric ...
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6 votes
0 answers
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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 ...
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2 votes
0 answers
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Categorical dimension and formal codegrees

Let $\mathcal{C}$ be a complex fusion category. If it admits a pivotal structure $a$ then by [1, Proposition 4.7.12], $\dim_a$ induces a character $\chi$ on the Grothendieck ring $Gr(\mathcal{C})$, of ...
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5 votes
0 answers
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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 ...
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4 votes
0 answers
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Are PD-, $\lambda$-, $\psi$-, and $\delta$-rings monoids in a monoidal category?

$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category: A monoid ...
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9 votes
1 answer
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Are differential rings monoids in a monoidal category?

$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category: A monoid ...
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5 votes
1 answer
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Is there a universal way to construct a bimonoidal category structure on $\mathsf{PSh}(\mathcal{C})$ from such a structure on $\mathcal{C}$?

The Day convolution monoidal category structure $(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$ on the category of presheaves of a monoidal category $(\mathcal{C},\...
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4 votes
1 answer
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Day convolution for bimonoidal categories

Semiring categories, also called rig categories or bimonoidal categories, are pseudomonoids in the symmetric monoidal bicategory $(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})$¹. These are a ...
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3 votes
1 answer
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Is Set (collectively) complete for Cartesian categories?

This paper says that $FinVect_k$ is collectively complete for traced symmetric monoidal categories, in the sense that given distinct arrows in the free traced SMC (over some generating monoidal ...
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Classification of biclosed monoidal structures on the $2$-category of $2$-categories

This paper proves that the category of small categories and functors between them admits exactly two monoidal biclosed categories: the cartesian tensor product and the funny tensor product. Is there a ...
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7 votes
1 answer
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Is $\oplus$ the only monoidal structure on the simplex category?

Simplicial sets are presheaves on the simplex category $\Delta$, while augmented simplicial sets are presheaves on $\Delta_+$, the augmented simplex category. Because Day convolution allows us to ...
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8 votes
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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 ...
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5 votes
1 answer
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End formulas for sets of monoidal natural transformations

Perhaps the characteristic feature of the theory of ends is that they are extremely useful for computing sets of transformations between two functors. For example, one has the formulas \begin{align*} \...
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4 votes
1 answer
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Link invariants from modular categories (strictification and computation)

By the theory of Reshetikhin and Turaev, a modular tensor category $C$ gives rise to a link invariant. While $C$ is strict as a monoidal category (e.g. $\mathbb{Fib}$), calculating the link can be ...
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10 votes
1 answer
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Taking the category of sheaves is symmetric monoidal

Let $M$ and $N$ be topological spaces. Let $\operatorname{Sh}(M)$ denote the presentable $\infty$-category of space-valued sheaves on $M$. It seems to me that the equivalence $$\operatorname{Sh}(M) \...
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  • 475
4 votes
0 answers
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Smash products of pointed groupoids

The category of pointed sets $\mathsf{Sets}_*$ has a symmetric closed monoidal category structure $(\wedge,S^0)$, which, analogously to the symmetric monoidal $\infty$-category of pointed spaces, is (...
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4 votes
0 answers
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The fusion categories with a strict skeleton

We refer to the book Tensor Categories (by Etingof-Gelaki-Nikshych-Ostrik) for all the notions mentioned in this post. A fusion category is skeletal if two isomorphic objects are always equal. Every ...
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7 votes
2 answers
544 views

Fields in monoidal categories

We can speak of rings in monoidal categories, including also the non-Cartesian case. What about fields? Question 1: Definitions What are some possible notions of a (skew or commutative) field in a ...
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