# Questions tagged [monoidal-categories]

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor ⊗ : C × C → C which is associative up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism. (Def: http://en.m.wikipedia.org/wiki/Monoidal_category)

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### Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
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### What's there to do in category theory?

I'm sure anyone who's heard of categories has also heard the classical "Well obviously there aren't any real theorems in category theory, it's much too general", or something in the likes of it. Now ...
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### What's a double category with one object?

Categories with one object are equivalent to monoids. $2$-categories with one object are equivalent to monoidal categories. Therefore, I am wondering whether double categories with one object are ...
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### Strongly unbiased symmetric monoidal category

Let $\mathcal{C}$ be a category. Define a strongly unbiased symmetric monoidal structure on $\mathcal{C}$ to be a rule which associates to every finite set $I$ a functor $\mathcal{C}^I \to \mathcal{C}$...
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### Generalization of analytic functors

A functor $F\colon \bf Sets\to Sets$ is said to be analytic if it results from the left Kan extension of a functor $f\colon \mathbf{Bij}(\mathbb N)\to \bf Sets$ (the "species" of the functors $F$) ...
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### Constructing free monoid from free semigroup

Given a monoidal category $(\mathcal{C}, \otimes, I)$ with coproducts, the free monoid on an object $A \in \mathcal{C}$ is usually constructed by first constructing the free pointed object on $A$, i.e....
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I'm interested in learning about monads and their relations to algebraic structures (as a generalization of universal algebra, if I understand well -correct me if not) . In the process of learning ...
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### Tannakian categories and differential Galois theory

I was taking a look at a result in Tamás Szamuely's Galois groups and fundamental groups. The following argument can be found right after the proof of lemma 6.6.7. Let $(K,\partial)$ be a ...
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### A (higher) categorical approach to representation theory

The representation theories of groups and Hopf algebras are very much alike. Taking the view point that both Hopf algebras and groups are Hopf monoids ("Hopf algebra objects") in their symmetric ...
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### When modular tensor categories are equivalent?

I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
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### Tensoring a connective chain complex with a simplicial set

Let $\mathrm{Ch}_{\geq 0}(R)$ be the category of chain complexes of $R$-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model ...
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### Mac Lane's Coherence Theorem: Why not just use the functors themselves?

I have a softball question on Mac Lane's proof. Suppose $B=\left ( B, \square , \alpha ,\rho ,\lambda \right )$ is a monoidal category. Fix $b\in B$. Define $W$, the (monoidal) category of binary ...
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### What's the difference between a cartesian monoidal category and a semicartesian monoidal category?

According to ncatlab: In a semicartesian monoidal category, any tensor product of objects $x \otimes y$ comes equipped with morphisms $$p_x : x \otimes y \to x$$ $$p_y : x \otimes y \to y$$...
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### On the definition of algebra .

Let $\mathscr{C}$ a monoidal category with monoidal product $A\circ B$. Is defined the bicategory of bimodules $Mod(\mathscr{C})$ on $\mathscr{C}$ (see [Gray] p. 46). Its objects are $\mathscr{C}$-...
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### Cartesian monoidal functors

Let $\mathcal{C}$ and $\mathcal{D}$ be categories with finite products, and consider them as monoidal categories in the obvious way. Every functor $\mathcal{C} \to \mathcal{D}$ can be canonically ...
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### Monoidal categories in which some tensor products of morphisms are equal

I have come across a certain monoidal category $(C,\otimes,I)$ (let us say it is strictly monoidal to simplify the notations, as usual it does not matter very much) which satisfies the following ...
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### For symmetric monoidal categories, the monoidal product is a monoidal functor

Has anyone got a reference for the following fact? If $\mathcal X$ is a symmetric monoidal category, then $\_\otimes\_\colon\mathcal X\times\mathcal X \to \mathcal X$ is a strong monoidal functor. ...
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### A monoidal category that preserves subobjects

Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be ...
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### Notation: why are categories enriched over $\mathcal V$?

In most references about enriched categories, $(\mathcal V, \otimes)$ is supposed to be a monoidal category and then $\mathcal V$-enriched categories are defined. Why is the letter $\mathcal V$ used ...
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### Does Projectiveness always imply flatness?

I know that a project module is always flat, deduced form the properties and abundance of free modules. I'm trying to figure out how essential role the free modules play in this result. So I'd like to ...
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### How to calculate braiding eigenvalues in a fusion category?

Statements like this are found in published articles: The context: Assume $\mathcal{C}$ is a complex fusion category (i.e. complex linear, finitely semisimple, monoidal, with duals, with simple ...
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### Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...
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### Examples of weak monoidal Quillen equivalences

Schwede-Shipley introduced the notion of weak monoidal Quillen equivalences between monoidal model categories in "Equivalences of monoidal model categories". Are there any examples of such Quillen ...
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### Adjunction of functors between monoidal categories induces adjunction between their categories of internal monoids

Here's a similar question, but I'm not sure if what I'll ask is already answered there. Let $(C,\odot), (D,\square)$ be monoidal categories. If $F \colon D \to C$ is left adjoint to $U \colon C \to D$...
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### Tensoring of the identity morphism of the monoidal unit

This is a rather elementary question about strict monoidal categories. Let $C$ be a monoidal category, $I \xrightarrow{id_I} I$ be the identity morphism of the unit object. How do I show that for any ...
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### Free monoidal category over a set II

In a previous question, a description of the free monoidal category over a set was given. Basically, it consists of formal expressions as objects and morphisms generated by associators and unitors, ...
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### How to represent convolution matrix as a string diagram?

I'm trying to represent common machine learning/deep learning concepts and operations in the form of string diagrams. The many categorical quantum mechanics papers by Coecke et al have given me a good ...
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### Monoidal equivalence is: A monoidal functor that is also an equivalence of categories

Let $(\mathcal C, \otimes, I)$ and $(\mathcal D, \otimes', I')$ be two monoidal categories. And let $(F,\phi, \psi)$ and $(G,\phi', \psi')$ be a monoidal functors from $\mathcal C$ to $\mathcal D$. ...
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### Uniqueness of quantizations

When studying quantum groups, in particular quantized universal enveloping algebras, people will tell you that such a quantization is in some sense unique. More specifically, you might hear that a ...
Assume $(C,\otimes,e)$ to be a strict abelian tensor category with additive tensor functor and $A$ an algebra object in $C$. Then we're able to define the category of $A$-right modules $C_A$ and the ...
Consider a coquasitriangular Hopf-algebra $(H,\mu,\eta,\Delta,\epsilon, S;r)$ over a field $\mathbb F$ with characteristic zero and the braided monoidal category $\mathcal M^H$ of $H$-right-comodules. ...