The monoidal-categories tag has no usage guidance.

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### About the closed structure on the modules of a monoidal closed symmetrical category

Let $(\mathscr{C}, \otimes , I)$ monoidal category, a monoid $(R, e_R, \mu )$ is a object $R \in \mathscr{C}$ with morphisms $e_R: I \to R$, $\mu: R \otimes R \to R$ with the well knowed unital ...

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### Two questions about commutative theories

Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...

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### The multiplicative system in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category. In the 1973 paper "Note on monoidal localisation" by Brian Day, the multiplicative system of morphism in $\mathcal{C}$ has been discussed. See also ...

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624 views

### Saavedra's Definition of Tannakian Category

I have been reading Deligne-Milne's Tannakian Categories and got to the point at the end of part 3 where they discuss what went wrong with Saavedra's definition. Motivated by their counterexample, ...

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### Braidings and Isomorphism Classes in a Monoidal Category

Let $X$ be an object in a monoidal category $({\cal C}, \otimes)$, and $\gamma:X \otimes X \to X \otimes X$ a braiding (that is to say a morphism in ${\cal C}$ from $X \otimes X$ to itself that ...

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449 views

### Iterating monoid categories

Let $(C, \otimes)$ be a symmetric monoidal category (maybe braided is also okay). Then the category $\text{Mon}(C)$ of monoid objects in $C$ is also a symmetric monoidal category with the same ...

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375 views

### String diagrams for (weak) monoidal categories

Hi,
In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:
where $i_x$ and $e_x$ are the unit ...

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453 views

### When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?

Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...

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### Are pivotal categories the algebras for a cartesian monad?

It seems to be "known" but not written down that the following are more-or-less equivalent:
...

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386 views

### Thompson's group F and monoidal categories

(This is a cross-post from MathSE, as someone remarked that the question would be more appropriate on MO)
Fiore and Leinster have proved that if $\mathcal{A}$ is a monoidal category freely generated ...

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231 views

### A question on triangle identities

It is well known that pentagon+triangle identity of type (a1b) implies "all diagrams commute" monoidal category, in particular triangle identities of type (1ab) and (ab1). My question is that whether ...

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668 views

### Localization of a symmetric monoidal category at a single morphism

Let $C$ be a symmetric monoidal category, and $f : x \to y$ be a morphism in $C$. I would like to construct the localization $C_f$ explicitly, which solves the universal property
...

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651 views

### What non-monoidal functors on monoidal categories are used “in nature”?

Background
For my PhD dissertation, I've developed a categorical generalization of many different systems of denotational semantics for light linear logic (LLL). I'd like to see if I can use this ...

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477 views

### Module categories over symmetric/braided monoidal categories

Given an algebraically closed field $k$ and a finitely generated commutative $k$-algebra $A$, all simple modules over $A$ are 1-dimensional
What is the analogous statement for symmetric monoidal ...

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148 views

### Extending braidings to tensor powers

Given a braiding $\Psi: X \otimes Y \to Y \otimes X$ for two objects $X,Y$ in a monoidal category, it seems reasonable to assume that $\Psi$ extends uniquely to a braiding $X^k \otimes Y^l \to Y^l ...

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532 views

### Hovey's unit axiom in monoidal model categories

Let $\mathcal{C}$ be a monoidal model category in the sense of Hovey's book. He assumes the following unit axiom not considered in other references (e.g. Schwede-Shipley): given a cofibrant ...

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### Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.

In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation:
Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every ...

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258 views

### Reshetikhin-Turaev and links with a distinguished component

Hi,
This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question.
Let $T$ be the category whose objects are ...

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256 views

### Definition of enriched caterories or internal homs without using monoidal categories.

I know this question may seem nonsensical at first but let me exlain what i have in mind:
In enriched category theory we define categories enriched over a monoidal category $(\mathcal{V},\otimes, ...

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429 views

### Does the dual of an object with trivial symmetry also have trivial symmetry?

Let $C$ be a symmetric monoidal category. I am interested in objects $X \in C$ such that the symmetry
$S_{X,X} : X \otimes X \cong X \otimes X$
is equal to the identity. There are many examples of ...

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### Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...

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734 views

### Gamma spaces and monoidal categories

In his paper "Categories and cohomology theories" Graeme Segal gives examples how to construct a Gamma category and therefore also a Gamma space from a strict monoidal category like finite chain ...

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### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...

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1k views

### Understanding Penrose diagrammatical notation

I arrived to Penrose's paper Applications of negative dimensional Tensors after reading some bits of Baez's Prehistory (link) and the first two chapters of Turaev's Quantum invariants of knots and ...

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### Show that duality functor is anti-monoidal

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects ...

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### When is the cofibrant replacement of a product the product of the cofibrant replacements?

I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE ...

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261 views

### Ends as a “cotrace” operation on profunctors

As mentioned here, there is a trace operation on the monoidal category of profunctors given by taking coends: for any profunctor $F : A\times X \nrightarrow B \times X$, there is a profunctor $Tr^X(F) ...

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925 views

### Exterior powers in tensor categories

Let $\mathcal{C}$ be a cocomplete $R$-linear tensor category. Many notions of commutative algebra can be internalized to $\mathcal{C}$. For example a commutative algebra is an object $A$ in ...

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### Action on tensor power and “element notation” in monoidal categories

Let $C$ be a symmetric monoidal category. Fix an object $X$, let $S$ denote the symmetry $X \otimes X \to X \otimes X$. Also define $X^{\otimes n}$ by induction on $n$: $X^{\otimes 0} = 1$, ...

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### A simple example of a tame tensor functor

Let $C$ be a $R$-linear cocomplete tame abelian tensor category (see here for definitions) with unit $1$, then there is a unique $R$-linear cocontinuous tensor functor $F : \text{Mod}(R) \to C$, ...

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### representing tensor-C*-categories in BIM

Given a factor M (=von Neumann alg. with center ℂ), let us write BIM for the ⊗-C*-category of M-M-bimodules.
Which ⊗-C*-categories can one faithfully embed into BIM?
⓵ Are ...

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### Is the functor of divided powers a weakly monoidal functor?

Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every ...

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### Tame abelian tensor categories

In the article "Tannaka duality for geometric stacks" (arxiv, see nlab for a summary) Jacob Lurie introduced the notion of a tame abelian tensor category. An abelian tensor category is called tame if ...

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### Do plain functors out of monoidal categories factor into a nontrivial monoidal part and a plain part?

Given symmetric monoidal closed categories $C, D$, a symmetric monoidal (not closed!) functor $F:C\to D$ factors into two parts:
the first is a symmetric monoidal closed functor from $C$ to a ...

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539 views

### universal property of module categories internal to a tensor category

By a tensor category, I mean here a cocomplete $k$-linear symmetric tensor category, where $k$ is a fixed ground ring. Tensor functors are assumed to be $k$-linear and cocontinuous. I will denote the ...

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### Epimorphisms between line objects

Let $\mathcal{A}$ be a $k$-linear abelian symmetric tensor category with unit $\mathcal{O}_A$; here $k$ is a comm. ring. By that I assume implicitly that $\otimes$ is finitely cocontinuous in each ...

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### Does one of the hexagon identities imply the other one?

Suppose we have a monoidal category equipped with additional data that almost makes it a braided monoidal category except that only one of the hexagon identities
is satisfied.
Can we then prove the ...

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486 views

### Gerbes and Z-graded symmetric monoidal categories

Let me first recall a known fact. Suppose $X$ is a complex algebraic variety, $\mathscr{L}$ is a line bundle on $X$ and $L^\times$ is the total space of $\mathscr{L}$ with the zero section removed. ...

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### graded ring associated to a line bundle in a tensor category

Let $\mathcal{A}$ be an abelian tensor category with unit $\mathcal{O}$. An object $\mathcal{L}$ is called invertible or a line bundle if there is some $\mathcal{L}^{-1}$ such that $\mathcal{L} ...

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### Alternative definition of monoidal categories

My question is about monoidal categories. To motivate it, let me first recall something about group objects.
Assume you define a group object in a category $C$ with products by an object $G$ together ...

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334 views

### Is the box product of morphisms associative?

Suppose $(C,\otimes)$ is a symmetric monoidal finitely-cocomplete category such that $\otimes$ preserves colimits. Given two morphisms $a:A_1\to A_2$ and $b:B_1\to B_2$, define $a\Box b$ to be the ...

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### Symmetric monoidal structure on cosimplicial spaces

Is there a monoidal structure on the category $Spc^{\Delta}$ of cosimplicial spaces such that in the adjunction
$$
\Delta^{\bullet}\otimes-\colon Spc\leftrightarrows Spc^{\Delta}\colon Tot(-)
$$
the ...

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### The condition End(1) = k in Tannakian Categories

A neutral Tannakian category over a field $k$ is a rigid $k$-linear abelian tensor category
$\mathcal{C}$ whose unit $1$ satisfies $\mathrm{End}(1) \simeq k$, and is
moreover equipped with an exact ...

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### What is the proper name for “compact closed” multiplicative intuitionistic linear logic?

Multiplicative intuitionistic linear logic (MILL) has only multiplicative conjunction $\otimes$ and linear implication $\multimap$ as connectives. It has models in symmetric monoidal closed ...

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### From tensor algebras in monoidal categories to (commutative?) monoids

Let $D$ be a monoidal category (without the structure of a symmetric monoidal category), with unit object $Id$, and let $L$ be an invertible object in $D$, so that $L$ is dualizable and the pairing ...

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### A better way to compute the mapping spaces of the category of spans in an enriched tensored category?

Let X be a tensored and cotensored V-category, where V is a fixed complete, cocomplete, closed symmetric monoidal category.
Define $C:=Span(X)$ to be the category of spans in X (this is the functor ...

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### What is the right way to define the nerve of an unbiased monoidal category?

I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One ...

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### Skeleton of a braided monoidal category

Does every (lax) braided monoidal category have a braided monoidal skeleton? That is, I want to define a (lax) braided monoidal structure on a skeleton so that it is braided monoidal equivalent to ...

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### In which categories is every coalgebra a sum of its finite-dimensional subcoalgebras?

Let $\mathcal V$ be a reasonably nice category — I'm interested in the case when $\mathcal V$ is $\mathbb K$-linear for some field $\mathbb K$, abelian, and has all products and coproducts ...

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### 180˚ vs 360˚ Twists in String Diagrams for Ribbon Categories

Ribbon categories are braided monoidal categories with a twist or balance, $\theta_B:B\to B$, which is a natural transformation from the identity functor to itself. In the string diagram calculus for ...