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Questions tagged [monoidal-categories]

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14
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2answers
316 views

Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...
4
votes
1answer
148 views

Is the category of rational Lie algebras monoidal?

I hate to ask such a naive question, but here goes. Suppose $A$ and $B$ are rational Lie algebras, i.e. rational vector spaces together with a bracket. Then, $A\otimes_{\mathbb{Q}} B$ is a rational ...
4
votes
1answer
103 views

Tensor product of modules over a monoid in a monoidal category

Assume $\mathcal{C}$ is a monoidal category, with unit $I$. Given a monoid object $M$, I'd like to talk about modules over $M$, but couldn't find any reference. This might seem quite a stretch, but it ...
8
votes
2answers
209 views

“Closed bicategories”

I am interested in the following property that a bicategory may or may not have. Let $\mathbf{B}$ be a bicategory. Every one-morphism $f\colon x\rightarrow y$ defines a functor $\mathbf{B}(y,z)\...
2
votes
0answers
30 views

Homotopy monoidal structure from Hirsch Algebra

In the paper available: https://escholarship.org/uc/item/77r5k6cb The differential is defined on page 13. Is this the wrong differential? Where are the $E_{p,q}$ maps coming from the twisted tensor ...
11
votes
1answer
203 views

Definitions of enriched monoidal category

This question is about two definitions of enriched monoidal categories I have: Let $\mathcal{V}$ be a symmetric monoidal closed category. The first definition: a $\mathcal{V}$-enriched category $\...
2
votes
1answer
166 views

The category of Multisets and Spans: morphism composition and tensor product

I have been thinking about multisets for a while. These are sets where elements can repeat, so $S =\{ a,a,b,c,b\}$ is a multiset on the set $A = \{a,b,c\}$. I have also been looking into morphisms ...
6
votes
1answer
265 views

Modules over Hopf Algebras and $E_2$-algebras

Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf I was wondering if anybody knows of a nice relationship between ...
5
votes
1answer
181 views

Dualizable presheaves with respect to Day convolution

This question was posted on MSE and got very little attention, so I'm also posting it here. Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \...
4
votes
1answer
134 views

DGA for a general abelian category

Differential graded algebras dga-s are fundamental objects of study in homological algebra and category theory. On the nlab webpage, they are defined as follows: a dga is a monoid in the symmetric ...
9
votes
2answers
220 views

Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups

For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...
3
votes
1answer
112 views

Tannaka-Krein reconstruction and rigidity

Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have ...
8
votes
2answers
158 views

What is a true invariant of $G$-crossed braided fusion categories?

Definition. An invariant of a (spherical) fusion category with extra structure is a number or a set or tuple of numbers preserved under (appropriate) equivalences. (Spherical) fusion categories have ...
9
votes
2answers
252 views

Balls in Lawvere metric spaces

Let $V$ be the monoidal category $[0,\infty)$ (as a poset) with $+$ and $0$. Lawvere shows that $V$-enriched categories are a more natural generalisation of the notion of a metric space (note no ...
7
votes
1answer
111 views

Bicategory of bimodules over internal monoids

In a monoidal category $(C,\otimes)$ one can consider internal monoids and bimodules over these monoids (provided some additional requirements like existence of coequalizers hold). These monoids ...
12
votes
2answers
775 views

What is a tensor category?

A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term tensor category is also used, and seems to mean a monoidal category with more ...
11
votes
3answers
263 views

Unbiased Hopf algebras

In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...
3
votes
1answer
115 views

Is the category of enriched operads (co)complete?

Let $V$ be a symmetric monoidal category which is complete and cocomplete. Is the category of small symmetric colored $V$-enriched operads complete and cocomplete? If $V$ is presentable, is it ...
9
votes
1answer
176 views

Elementary equivalence of monoidal categories =?

Recall that, in model theory, two models $M_1$ and $M_2$ of the same signature are elementary equivalent if $ M_1 \models \phi \Leftrightarrow M_2 \models \phi $ for every first order formula $\phi$ ...
1
vote
1answer
119 views

Abstracting the properties of the category $\frak{g}$-modules

Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^*$, we know that the decomposition of $V \otimes V^*$ ...
5
votes
2answers
194 views

Enrichments vs Internal homs

Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor $$ X \otimes -: \cal{C} \to \cal{C}, $$ for ...
8
votes
4answers
443 views

The dual of a dual in a rigid tensor category

For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{*})^*$, for $*$ denoting dual? If so, what is a good example.
8
votes
2answers
284 views

On functors preserving monoid objects

If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids: ...
10
votes
0answers
188 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 ...
6
votes
1answer
179 views

name for monoids inducing bimonoids in Rel?

Let Rel be the category of sets and relations, which is a (compact closed) symmetric monoidal category under the cartesian product of sets. We write $A \nrightarrow B$ to indicate a relation from $A$ ...
8
votes
0answers
198 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 ...
5
votes
2answers
188 views

Does the Day convolution induce the structure of a bimonoidal category on $Fun(C,D)$?

Let $\mathcal{C}$ and $\mathcal{D}$ be symmetric monoidal categories and assume that the symmetric monoidal product $\otimes_{\mathcal{D}}$ on $\mathcal{D}$ preserves colimits in both variables. Then ...
7
votes
2answers
249 views

MacLane coherence theorem for “monoidal” category without 1

MacLane's coherence theorem for a monoidal category states that once the associators for 4-fold products are compatible (i.e., the pentagon axiom holds), it holds for n-fold products, so I can bracket ...
11
votes
1answer
250 views

What do you call $C$ if $[D,C] = D^\vee \otimes C$ for all $D$?

This is different from $C$ being dualizable ($[C,D] = C^\vee \otimes D$). (EDIT: It turns out to be the same -- see Mike Shulman's answer!) But for example, if $C$ is a locally free sheaf of finite ...
7
votes
1answer
176 views

On the isomorphism problem of enveloping algebras

Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \...
5
votes
1answer
76 views

A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module

For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \...
2
votes
1answer
140 views

A non-monoidal functor that respects fusion rules

Let $(\cal{C},\otimes)$ and $(\cal{D},\odot)$ be two monoidal categories. Moreover, assume that $\cal{C}$ and $\cal{D}$ are abelian and semisimple. Let $X,Y$ be two simple objects in $\cal{C}$, and ...
0
votes
0answers
58 views

What is the name for a natural transformation that has both lax and oplax monoidal properties?

Let $\mathcal C,\mathcal D,\mathcal E$ be monoidal categories, let $g$ be an oplax monoidal functor from $\mathcal C$ to $\mathcal D$ and let $G$ be a lax monoidal functor from $\mathcal D$ to $\...
2
votes
0answers
107 views

Quantum invariant: The canonical $2$-tensor

In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...
3
votes
0answers
176 views

What is the name of this construction on monoidal categories?

$\newcommand{\C}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\F}{\mathcal F} \renewcommand{\H}{\mathcal H} \newcommand{\from}{\colon} \newcommand{\tensor}{\otimes} \require{AMScd}$ Given ...
39
votes
2answers
3k views

The two ways Feynman diagrams appear in mathematics

I've heard about two ways mathematicians describe Feynman diagrams: They can be seen as "string diagrams" describing various type of arrows (and/or compositions operations on them) in a monoidal ...
5
votes
0answers
126 views

Examples/Papers where a monoidal structure restricts to a promonoidal structure on a dense full subcategory

In Brian Day's thesis, he gives a definition of a pro-monoidal structure based on a particular motivating example: Suppose $M:\cal{A}\hookrightarrow \cal{B}$ is the inclusion of a small full ...
1
vote
0answers
71 views

Detecting skew-primitives in representation categories

Suppose $H$ and $H'$ are two (possibly infinity dimensional) Hopf algebras which are not isomorphic as Hopf algebras, but are isomorphic as algebras. More specifically they are not isomorphic as Hopf ...
5
votes
1answer
202 views

Internal Hom of Deligne' tensor product

I read the following statement (equation 22) in "Monoidal 2-structure of bimodule categories" by Justin Greenough: Let $\mathcal{C}$ be a finite tensor category (abelian k-linear rigid monoidal ...
8
votes
2answers
339 views

Special $\Gamma$-categories and symmetric monoidal categories

Let $\Gamma^{op}$ be the category of finite pointed sets. A special $\Gamma$-category is a functor $Y:\Gamma^{op}\to Cat$ such that the canonical maps $Y[n]\to Y[1]^n$, called Segal maps, are ...
2
votes
0answers
126 views

How does the relative Drinfeld center interact with the relative Deligne tensor product?

Let $\mathcal{C}$ be a fusion category, and $\mathcal{M}, \mathcal{N}$ semisimple $(\mathcal{C}, \mathcal{C})$-bimodule categories. The left $\mathcal{C}$-action is denoted as $- \triangleright - \...
4
votes
0answers
107 views

Category of (co)commutative Hopf monoids in an exact category

I'm transferring this question over from SE, since it didn't get much attention over there. Let $(C, \otimes)$ be an exact monoidal category, and let $H(C)$ be the category of cocommutative and ...
3
votes
1answer
267 views

How to understand the Deligne' tensor product of finite abelian category

In the sec 1.11. "Delignes' tensor product of locally finite abelian categories" of the book "Tensor Categories" of EGNO, the deligne's tensor product $C \boxtimes D$ of two k-linear locally finite ...
9
votes
1answer
264 views

Uniqueness of dualizing objects

One definition of (symmetric) star-autonomous category is as a closed symmetric monoidal category $(C,\otimes,I,\multimap)$ equipped with an object $\bot$ such that all double-dualization maps $A \to (...
6
votes
2answers
232 views

Graded rings with compatible S_n actions

Does the following mathematical gadget have a standard name? Let $R$ be an $\mathbb{N}$-graded ring together with an $S_n$ action on each $R_n$ which are compatible in the following sense. Let $i:...
13
votes
1answer
259 views

The Chu-construction and the Int-construction

The Chu construction is a way of building a star-autonomous category $\mathrm{Chu}(C,\bot)$ from any closed monoidal category $C$ with pullbacks and a choice of an object $\bot\in C$ to become the ...
10
votes
1answer
175 views

Skeletal monoidal categories with strict units

I believe that every skeletal monoidal category is monoidally equivalent to a skeletal monoidal category with strict units. Does anybody know a reference for this fact in the literature?
14
votes
1answer
219 views

String diagrams for bimonoidal categories (a.k.a. rig categories)?

I'm having some fun playing around with string diagrams for monoidal categories, expressing familiar constructions from Riemannian geometry and linear algebra in terms of elegant string diagrams. I'...
3
votes
0answers
124 views

Can monoids of “continuous words” be realized as initial monoid objects?

Whenever $X$ is a set, write $X^*$ for the monoid freely generated by $X$. The elements of $X$ are, of course, words in the letters $X$. When $X$ is finite, there also seems to be a great many ...
6
votes
0answers
117 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 ...