# Questions tagged [monoidal-categories]

The monoidal-categories tag has no usage guidance.

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### 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**

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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**

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**1**answer

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 ...

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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)\...

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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**

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**1**answer

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**

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**1**answer

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**

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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**

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**1**answer

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}, \...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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**

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**1**answer

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 $\phi$ ...

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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 $V \otimes V^*$ ...

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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**

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**4**answers

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.

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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:
...

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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 ...

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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 $A$ ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 \...

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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**

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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 ...

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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 $\...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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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 ...

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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 ...

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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 ...

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### 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 - \...

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### 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 ...

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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 ...

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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 (...

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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:...

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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 ...

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### 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?

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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'...

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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 ...

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### 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 ...