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/...

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Monoidal structure on the opposite category of a symmetric monoidal category

I have two questions that confuse me a lot. First of all, can someone state the exact way in which, given a symmetric monoidal category $\mathcal{C}$, the opposite category $\mathcal{C}^{\text{op}}$ ...
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3answers
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Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
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Is the tensor algebra functor a strong monoidal functor?

Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
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Balanced Tensor Product of Module Categories

Let $C$ be a $k$-linear ($Vect_k$-enriched) monoidal category and consider the 2-category $Mod_{C}$ of $k$-linear $(C,C)$-bimodule categories in the sense of Ostrik (https://arxiv.org/abs/math/0111139....
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Examples of asymmetrically braided monoid

From nCatlab https://ncatlab.org/nlab/show/braiding : Any braided monoidal category has a natural isomorphism $$B_{x,y} \;\colon\; x \otimes y \to y \otimes x $$ called the braiding. ...
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A question regarding monoidal closed categories

If a category $\mathcal{C}$ is (symmetric) monoidal closed, is the opposite category $\mathcal{C}^{\text{op}}$ also monoidal closed? It is not clear to me whether by dualising the natural bijection $...
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60 views

Completeness of 2-category of Monoidal Categories

Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?
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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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38 views

Can tensor abelian categories always be embedded into the category of modules?

Let $(\mathcal A, +,\otimes,I)$ a small symmetric monoidal abelian category. I know that $\mathcal A$ can be embedded into the category of $R$-module for a certain ring $R$. But can we make such ...
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58 views

A question about monoidal categories

I am learning about monoidal categories and I am a bit confused about the following: Suppose $(A,\otimes,I)$ is a monoidal category. What can be said about the opposite $A^{\text{op}}$? Is it ...
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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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The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category $\...
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How to call a “non-strict” monoidal category?

A monoidal category is a category $\mathsf{C}$ equipped with a bifunctor $\otimes : \mathsf{C} \times \mathsf{C} \to \mathsf{C}$, a unit object, an associator, and right and left unitors satisfying a ...
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1answer
16 views

Enriching categories of simplicial objects

Let $C$ be a cocomplete category and $Simp(C)=C^{\triangle^{op}}$ the category of simplicial objects in $C$. I want to show that $Simp(C)$ is simplicially enriched but I don't understand how the ...
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2answers
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Epimorphisms and faithful functors in a rigid abelian tensor category

Let $\mathsf{C}$ be a rigid abelian tensor category in the sense of Deligne and Milne's notes (p.9). Let $\mathbf{1}$ denote the identity object in $\mathsf{C}$ with respect to $\otimes$. The abelian ...
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25 views

Adjointness of internal contravariant Hom in symmetric monoidal categories.

Let consider a closed symmetric monoidal cateogry, $\mathscr C,\otimes$, with adjunction $(X\otimes-)\dashv\mathrm{Hom}(X,-)$ for all objects $X$. The following isomorphis, valid in categories such ...
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Functor between Category and its Free Strict Monoidal Category

Let $C$ be a category and let $\sum(C)$ denote the free strict monoidal category over $C$. According to Wikipedia, the operation $\sum: C\rightarrow\sum(C)$ extends to a 2-monad on $Cat$. Can someone ...
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Rings that cannot be representations rings

Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo ...
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If two categories are equivalent and the one is monoidal, is the other monoidal too?

Pretty much what I ask in the title. Let $\mathcal{C},\mathcal{D}$ be two categories and suppose there exists a fully faithful, surjective-on-objects functor $F:\mathcal{C}\to\mathcal{D}$ (so that $\...
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25 views

Left and right endomorphisms in monoidal categories

Let consider a monoidal category $\mathscr C, \otimes$ with unit $I$ and unitors $\lambda_X:I\otimes X\to X$ and $\varrho_X:X\otimes I\to X$. Each endomorphism of the unit $a:I\to I$ induces, on each ...
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48 views

Coherence diagrams for monoidal categories which have underlying sets are “automatically” natural?

I am just getting into monoidal categories, and first I am verifying at $FdVect$ is one, the monoidal operation being the usual vector-space tensor product. (Which I also just learned, so I may be ...
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44 views

Free monoidal category over a set

From nLab's article on coherence theorems, there seems to be a notion of free monoidal category over a set $S$. I guess this corresponds to the left adjoint to the functor $Ob : MonCat \to Set$ which ...
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2answers
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Why are invertible objects reflexive in a tensor category?

I am reading Deligne and Milne's notes on Tannakian categories. I'm only just getting used to the idea of an abstract tensor category, and I've encountered a very believable statement that I'm ...
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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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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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Category of finitely presented $R$-algebras cartesian closed?

On page 26 of these notes, in the paragraph between formulas $(63)$ and $(64)$, the author says the category of finitely presented $R$-algebras is cartesian closed. I thought this category was ...
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120 views

Does $f\otimes \operatorname{Id} = \operatorname{Id}$ imply $f= \operatorname{Id}$?

Let $R$ be a commutative ring, and $X$ an $R$-module. If an $R$-endomorphism of $X$ satisfies $f\otimes \operatorname{Id}_X = \operatorname{Id}_{X\otimes X}$, is it true that $f=\operatorname{Id}_X$ ? ...
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Three different definitions of Modular Tensor Categories

I have found three different definitions of Modular Tensor Categories. I want to know if anybody can give a sketch of proof for their equivalences (some parts are easy of course) A Modular Tensor ...
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61 views

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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3answers
172 views

Day convolution intuition

In the nLab, Day convolution is introduced as a generalisation of convolution of complex-valued functions, but I'm wondering how exactly to understand this. I can (just about) parse the definitions, ...
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1answer
30 views

Modular tensor category and the pivotal or shperical condition

I have two related questions: 1) Is a modular tensor category always pivotal? 2) Is a modular tensor category always spherical?
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Is the braid category biclosed and bicomplete?

Let $\mathcal{B}$ be the braid category, as in Categories for the Working Mathematician §XI.4 p.262 (objects are natural numbers and morphisms are the braids $n\to n$). Then this can be given a ...
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Recovering $\mathsf{Ab}$ from $\mathbb{Z}$-mod in a closed symmetric monoidal category

Given the usual definition of a module over a ring it is trivial to show that a $\mathbb{Z}$-module is an abelian group (it is just by definition). But my question concerns recovering this idea in a ...
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Categorification of algebra structures

This might be a bit of a soft question. Take a $\mathbb{C}$-linear category. Form the complex vector space spanned by its objecs modulo exact sequences. This construction is, as far as I know, the ...
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In what sense right dual and braiding structure respect the tensor product structure in a monoidal category?

Throughout let $(\mathscr{C}, \otimes, \mathbf{1})$ be a monoidal category (I suppressed unitors and associators for simplicity). The usual definition a rigid monoidal category is done in two steps: ...
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What is a morphism of Tannakian categories?

In this question, a Tannakian category over $k$ is a $k$-linear rigid symmetric monoidal tensor category, with the property that it has a fibre functor to $\mathbf{Vect}_\ell$ for some field extension ...
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1answer
72 views

initial algebra and free monoid

I am missing something to prove that the initial algebra $A^*=\mu x. Fx$ of the functor $FX=I+A \otimes X $ is the free monoid in a monoidal category. Here's one start Summing up I can build 2 ...
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1answer
28 views

closed monoidal posets

Let $X$ be a set, regarded as discrete category. if $X$ has structure of closed monoidal category $(X,\cdot,e)$, then it is easy to show that $X$ is a group: since all the morphisms are identities, ...
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1answer
106 views

$\mathcal{V}$-naturality in enriched category theory

Let $\mathcal{V}$ be a monoidal category, in section 1.2 of "Basic concepts of enriched category theory" (http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf) Max Kelly introduces the terms "$\...
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1answer
77 views

Extended Topological Quantum Field Theory, (ETQFT) basics ..

What is the functorial (categorical) definition of a TQFT (Topological Quantum Field Theory), which Jacob Lurie "had extended", for his ETQFT ? Actually I just need to know what are basic tools, to ...
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1answer
41 views

When a monoidal category is equivalent to its center

The notion of the center of a monoidal category categorifies that of the center of a monoid. Similarly, the notion of a braided monoidal category is a categorification of that of a commutative monoid. ...
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When is the tensor unit a generator?

Let $\mathcal{C}$ be a monoidal category with the tensor unit $I$. Then there is a "forgetful" functor from it to $Sets$: \begin{equation} \mathrm{Hom}(I,-). \end{equation} But in general, this ...
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1answer
35 views

Free commutative ring functor

The free commutative ring on a set $X$ is the polynomial ring with variables the elements of $X$. This polynomial ring is the free (additive) abelian group on the free (multiplicative) abelian monoid ...
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1answer
31 views

Understanding dual (noncommutative) modules / vector spaces functorially

For vector spaces over a fixed field $\Bbbk$, there's a natural transformation from the identity functor to the double dual functor. I think here's a way to construct it. Start from the identity arrow ...
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Tortile Yang-Baxter operators and Tortile Monoidal Categories

I am trying to understand Proposition 1 of the paper "Tortile Yang-Baxter operators in tensor categories" (By André Joyal & Ross Street), but I did not understand the second line of the proof. In ...
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1answer
129 views

algebra of endomorphisms of a functor

Let $C$ be a finite $k$-linear abelian category. Let $F:C \to \mathrm{Vec}$ be exact faithful functor to the category of (finite dimensional (need?)) vector spaces. Let $\mathrm{End}(F)=\mathrm{Nat}(...
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1answer
52 views

Spelling out uniqueness in a rigid category

As has been discussed in other posts, the dual of an object in a rigid category is unique up to unique isomorphism. As highlighted here, this does not mean that, for any two duals $(X^*,\epsilon,\nu)$ ...
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What natural monoidal structure and braiding exists on the category of modules of the convolution algebra of an action groupoid?

Let $S$ be a set with an action $\triangleright$ of a finite group $G$. The action groupoid $S // G$ has as objects the set $S$, and the morphisms from $s_1$ to $s_2$ are just the $g \in G$ that ...
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1answer
81 views

Looking for intuition and/or insight regarding the “into-internalization principle” in category theory.

Observation 0. Suppose $I$ is a set. Then the category $\mathbf{Set}^I$ can be explicitly described as the category whose objects are as follows: an object is a set $S$ equipped with a function $$I \...
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Tensor products and morphisms

Let $C$ be semisimple category with simple objects $X_1, \dots, X_r$. Suppose we have a fusion relation $X_i\otimes X_j =\bigoplus_{l=1}^r N_{ij}^l X_l$. Let $m\in \mathbb{N}$ and let $g:mX_j \to ...