Questions tagged [monoidal-categories]

A monoidal category, also called a tensor category, is a category $\mathcal{C}$ equipped with a bifunctor $\otimes\colon \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ which is associative up to a natural isomorphism, and an object $\mathbb{1}$ which is both a left and right identity for $\otimes$ up to a natural isomorphism.

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Is Set coclosed?

Background: $\mathbf{Set}$ is a (cartesian) closed monoidal category, so we have the natural tensor-hom adjunction $\text{Hom}(X\times Y,Z)\cong\text{Hom}(X,\text{Hom}(Y,Z))$ for sets $X,Y,Z$. ...
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Associahedron, but with swaps

The associahedron has edges of the form $a(bc)\rightarrow (ab)c.$ But I also want to include the possibility of swapping adjacent entries by doing operations like $a(bc) \rightarrow a(cb).$ I was ...
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Does this implication hold in a ribbon fusion category?

Let $\mathcal{C}$ be a ribbon fusion category in the sense of e.g. Bakalov-Kirillov Lecture Notes on tensor categories and modular functors. The notion of the $S$-matrix is of course well-known. ...
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Is the unit compact in a closed monoidal (co)complete category?

Let $\mathcal{M}$ be a complete and cocomplete closed symmetric monoidal category, with unit $I.$ Does the functor $\text{Hom}_{\mathcal{M}}(I,-)$ always preserve filtered colimits? A reference or a ...
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The definition of a free monoidal category

I want to understand what a free monoidal category is, over a signature $\Sigma,$ as described just before Theorem 2.3 of Selinger. In particular, I am hoping to get a more explicit description of the ...
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When is a left or right invertible morphism between isomorphic objects an isomorphism?

I just happened upon the age-old question in the title. We all know that an injection or surjection between e.g. finite-dimensional vector spaces of the same dimension or finite sets of the same ...
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Why a monoidal category with only one object is a monoid in the category of monoids?

I'm reading Chapter 4 in Steve Awodey's Category Theory and it mentions A discrete monoidal category, that is, one with a discrete underlying category, is obviously just a regular monoid (in $Sets$), ...
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Self-dual object in tensor triangulated categories which are not strongly dualizable

I am working in a closed symmetric tensor triangulated category. This is a triangulated category $\mathcal{T}$ admitting a symmetric monoidal structure with tensor product $\otimes$ which is closed. ...
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What makes every strong monad on a certain category be a monoidal functor?

A concept named Monad is used a lot in functional programming. And in spite their definition is not completely same with the definition of monad in category theory, as I know, Monad on a programming ...
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Intuition for definition of monoid in Category theory

I struggle real had trying to understand the definition of monoid in Category theory. At the first glance, the definition of monoid seems nothing but the definition in abstract algebra, but when I try ...
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Left inverse in monoid, left dual in monoidal category, and uniqueness

The notion of monoidal category is a categorification of the notion of monoid. If $M$ is a monoid, consider the monoidal category $Vec_M$ (of $M$-graded vector spaces over a field $\mathbb{k}$). If an ...
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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 have an example of a 2-cell ...
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Symmetric Frobenius structure. [closed]

I am new to the Frobenius Algebra course. One of my textbook exercises ask to prove: Show that every finite-dimensional simple algebra admits a symmetric Frobenius structure. I have no idea how to ...
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A matrix ring over $\mathbb{C}$ is also a Frobenius algebra over $\mathbb{R}$

I am new to the Frobenius Algebra course. One of my textbook exercises ask to prove: A matrix ring over $\mathbb{C}$ is also a Frobenius algebra over $\mathbb{R}$ My attempt: I know that I can give a ...
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Are there adjoint functors that don't play nicely with internal homs?

Let $\mathcal{C}$ be symmetric monoidal closed, with tensor product $- \otimes -$ and internal hom $[-,-]$. In this case, we know that the tensor-hom adjunction internalizes, and $[X \otimes Y, Z] \...
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Are the left and right unitor maps of the unit object in a monoidal category the same?

Suppose we have a monoidal category $\mathbb{C}$, with monoidal product $\otimes$, monoidal unit $I$, left unitor components $I \otimes A \overset{\lambda_A}{\rightarrow} A$ and right unitor ...
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definition of "closed monoidal functor"

A (strong) monoidal functor $F:C\to C'$ between monoidal categories is a functor equipped with natural isomorphisms $$ F_0:I'\cong F(I) $$ $$ (F_2)_{A,B}:F(A)\otimes'F(B)\cong F(A\otimes B) $$ such ...
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Effectus theory and the Giry monad

Is the Kleisli category of the Giry monad a monoidal effectus with copiers, in the sense of Definition 70 from An Introduction to Effectus Theory ? The fact that this category is an effectus is ...
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Do comonoids in the closed monoidal category of topological spaces with the separate continuity product form a "nice" category of spaces?

The category of topological spaces $\mathbf{Top}$ can be made into a multicategory where maps $X_1,X_2,...,X_n \to Y$ are the maps $X_1 \times X_2 \times ... \times X_n \to Y$ that are continuous in ...
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Higher arity monoidal products

Suppose we have a monoidal category $\mathbb{C}$ with monoidal product $\otimes.$ The $\otimes$ operation gives us binary monoidal products, so we would have to build up an expression for a product of ...
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How is the morphism of composition in the enriched category of modules constructed?

In April, I asked the question of how the structure of the enriched category is introduced into the category $_AV$ of modules over a given monoid $A$ in a closed monoidal category $V$: if we consider ...
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Coherence theorem in a lax monoidal category without unit

I am currently trying to understand the coherence theorem in monoidal categories. I would like to understand (in terms similar to those used in MacLane's theorem 3.1 in https://inspirehep.net/files/...
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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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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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Relation between monoidal category and universal property of the tensor product?

The product of a monoidal category is called the tensor product. But looking at the conditions for it here, they seem very different from the universal property of the tensor product. Is there a ...
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Concretely calculating the Day convolution of $\text{Cat}$-valued presheaves on a monoidal category

Summary Let $(\mathcal{C}, \otimes, e)$ be a monoidal category, and consider the category $[\mathcal{C}^\text{op}, \text{Cat}]$ of $\text{Cat}$-valued presheaves and natural transformations. I want to:...
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What are the dualizable objects in the category of Hilbert spaces?

Let $\mathbf{Hilb}$ be the category of Hilbert spaces and continuous linear maps. Turn it into a symmetric monoidal category using the tensor product of Hilbert spaces. What are the dualizable objects?...
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Square of unit in a monoidal category

Suppose we have a monoidal category $(C,\otimes,I)$ with left and right unitor being $\lambda$ and $\rho$. They yield two morphisms $\lambda_I,\rho_I:I\otimes I\to I$. It seems to me that both ...
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When is a monoidal effectus a distributive monoidal category?

The definition of a monoidal effectus is given by Definition 62 of Cho et. al.. The definition of a distributive monoidal category is given in ncatlab. My question is when is a monoidal effectus a ...
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Coherence theorem for monoidal category with strong endofunctor

I am interested in a reference to a Mac Lane-type coherence theorem as follows. I have a symmetric monoidal category $C$ endowed with an endofunctor $\square$ which is a strong monoidal functor, i.e. ...
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$\bf M-Set$ is cartesian closed

I must show that, given a monoid $M$, the category $\bf \text{M-Set}$ is cartesian closed. Take a $M\text{-Set}$ $X$, and call $-^X$ the right adjoint of $-\times X$. For any $M\text{-Set}$ $Y$, $Y^X$ ...
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Does it make sense to ask what the category of arrows is for monoidal categories?

The arrow category $\text{Arr}(C)$ of a category $C$ has morphisms as objects and commuting squares as morphisms. Is the arrow category of monoidal categories significantly different? My thoughts ...
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Pushout of diagram of a Monoidal Product Category and it's Projection Functors

Suppose $M$ and $N$ are Monoidal Categories and let $M\times{N}$ denote the associated product category. $M\times{N}$ comes equipped with two natural projection Functors $\pi_{M}:M\times{N}\rightarrow{...
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Distributive monoidal categories

Can anybody point me towards a paper where distributive monoidal categories are defined properly ? I have seen them mentioned in nLab, but I wanted to check that nLab hasn't omitted any coherence ...
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Definition of planar isotopy of diagrams in a monoidal category

I am reading this paper about graphical language in categories and at page 11 the author introduces this intuitive notion of planar isotopy for diagrams of a monoidal category. I can see the intuition ...
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Objects in Ind-category are filtered colimits of compact subobjects

In his paper 'Categories Tensorielles' in section 2.2 Deligne states that if in a tensor category $\mathcal{A}$ all objects are of finite length, then every object of the Ind-category $\text{Ind}\...
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What is a good category for probability theory?

I am searching for a good category to think about probability theory in, with arrows as something like stochastic maps. There are certain nice structural features I would like the category to have (...
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Does Set, with union, form a monoidal category?

I hope this question is not too naive. Is it possible to think of $\text{Set}$ (or some significant full subcategory of it) as a monoidal category with set theoretic union as the tensor product ($A \...
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A tensor category need not be isomorphic to a strict tensor category

I'm reading the book "Tensor categories" by Etingoff (and others). In remark 2.8.6 (posted below), it is claimed that the category $\mathcal{C}_G^\omega$ (defined in example 2.3.8, also ...
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Why do we need $End(I)=k$ for neutral Tannakian categories?

I have been reading Milne's book 'Basic Theory of Affine Group Schemes', and in particular the section on Tannaka duality for affine group schemes. The 'final' theorem of this section is displayed ...
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Free monoidal and free Markov categories

I hope this question is not too vague. I suppose one way to think about all the paths from one vertex to another within graph is to think about the free functor from graphs to categories, which is the ...
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Definition of rigid monoidal category generated by a category

A rigid monoidal category is a monoidal category where every object has both left and right duals. Is there a precise definition of rigid monoidal category generated by a category $C$? What would it ...
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Rig vs bimonoidal categories

I was wondering if rig categories (as discussed here ncatlab: rig category for example), are the same thing as bimonoidal categories (as discussed here Bimonoidal Structure of Probability Monads for ...
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Why does this diagram commute?

I am trying to understand the commutativity of diagram (2) of Proposition 3.11 from this paper. It should be a simple computation but I don't understand how they do it and when I try to do it myself I ...
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Why is the category of $\textit{right } A$-modules a $\textit{left } \mathcal{C}$-module category?

Let $C$ be a monoidal category and let $A$ be an algebra in $C$. Why is the category $Mod_C(A)$ of right $A$-modules a left $C$-module category? Take any $Y \in Mod_C(A)$ and any $X \in C$. We want ...
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Closed embedding of topological groups and representation category

If $\mathcal{T}_1,\mathcal{T}_2$ are two neutral Tannakian categories with fibre functors $\omega_1,\omega_2$ respectively. Then a morphism of neutral Tannakian categories $$F:\mathcal{T}_1\to \...
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2answers
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an infinite monoid that is not free monoid and does not contain any free monoid [duplicate]

Let $H$ generated by some generators $H=\langle h_1, \ldots, h_n\rangle$ $(n\gt 1)$. My question is whether there exists any monoid $H$ such that $H$, is infinite and $H$ is not a free monoid and $H$ ...
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Category of symmetric monoidal functors out of a symmetric monoidal category with duals form a groupoud

In Lurie's proof of the cobordism hypothesis, he uses the following fact: Let $C,D$ be a symmetric monoidal categories with duals, and $Z_1,Z_2\colon C\to D$ be symmetric monoidal functors, then any ...
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Is the category of modules over an abstract monoid enriched?

If $A$ is an algebra over $\mathbb C$, or, in other words, a monoid in the closed monoidal category $_{\mathbb C}\operatorname{Vect}$ of all vector spaces over $\mathbb C$, then, clearly, the category ...
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Compactly generated stable categories are dualizable.

Let $1-\operatorname{Cat}^{\operatorname{St,cocmpl}} _{\operatorname{cont}}$ denote the ($\infty$-)category of stable cocomplete ($\infty$-)categories. The Ind-completion $\operatorname{Ind}(C_0)$ of ...

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