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: ...

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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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21 views

All Ideals are Radical in Rigid Categories

I am reading Balmer's paper "Spectra, Spectra, Spectra" regarding the spectrum of tensor-triangulated categories. I think I am missing something obvious when he states that all ideals are radical as ...
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39 views

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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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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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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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 ...
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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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24 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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23 views

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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54 views

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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1answer
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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47 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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31 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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59 views

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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1answer
73 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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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 ...
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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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46 views

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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28 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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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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53 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
151 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

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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1answer
64 views

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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2answers
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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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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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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
70 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
39 views

Universality of tensor product from monoidal structure

As a follow-up to this previous question of mine, I'm trying to understand how to obtain tensor products from internal homs. I'm having a lot of difficulties and have found myself stuck already in ...
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1answer
27 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
103 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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39 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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14 views

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
34 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
121 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 ...
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1answer
29 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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16 views

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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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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152 views

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 ...
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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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1answer
80 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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32 views

A monoidal functor of rigid monoidal categories is an isomorphism

Let $C$ and $D$ be rigid monoidal categories and let $F_1, F_2: C\to D$ be monoidal functors. Then I want to show that if $\eta: F_1 \to F_2$ is a natural transformation, then $\eta$ is an ...
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1answer
49 views

When an equivalence is a monoidal equivalence?

I want to understand the followings from "Tensor Categories" by Etingof, Gelaki, Nikshych, and Ostrik. Remark 1.5.3 It is easy to show that if $F: C \to C'$ is an equivalence of monoidal ...
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1answer
44 views

Is every equivalence of monoidal categories a monoidal equivalence?

Let $C$ and $D$ be monoidal categories. Let $T:C \to D$ be a functor that gives the equivalence of $C$ and $D$ as just categories. My question is whether such an equivalence $T$ is always a monoidal ...
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1answer
79 views

Initial strict monoidal category

nlab provides a universal property of the cube category $\Box$. Definition. The cube category is the initial strict monoidal category $(M,\otimes,I)$ equipped with an object $int$ together with ...
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Trace in monoidal category acts multiplicatively

Every paper I've found claims this is true but nobody actually proves it. Let $(C,\otimes)$ be a strict monoidal category with duals and for every object $a\in C$ an evaluation $\epsilon_a:a \otimes ...
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55 views

Connecting physical tensors to mathematical tensors

I feel like I (maybe) understand the mathematical /algebraic perspective on tensors, for example as described in Wikipedia/Monoidal Category. You need a (simultaneously left & right) unit and an ...
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1answer
31 views

If $\mathcal{C}$ is a monoidal, $R$-linear category, what does the notation $\mathcal{C}\otimes_R\bar{R}$ mean, if $\bar{R}$ a quotient of $R$?

Suppose $\mathcal{C}$ is a strict, monoidal $R$-linear category, where $R$ is some commutative ring. If $\bar{R}$ is a quotient ring of $R$, what does the notation $\mathcal{C}\otimes_R\bar{R}$ mean? ...
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1answer
154 views

Going from Closed Categories to Monoidal Categories

EDIT: I trimmed down the exposition a bit. I really just wanted everyone to know what my approach has been, but what I had was a bit bloated. Suppose we have a closed category $V$ as defined here, ...
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61 views

The category of elements, enrichment, and weighted limits

Every so often, when reading notes online or skimming through books, the category of elements and the Grothendieck construction pop up. I don't know anything about the Grothendieck construction, and I ...