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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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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61 views

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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54 views

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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63 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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25 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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96 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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36 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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29 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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25 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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112 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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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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78 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 ...
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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
44 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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41 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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13 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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71 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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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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26 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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17 views

What is the map $\mathrm{Nat}(F_1,F_2)\times\mathrm{Nat}(G_1,G_2)\to\mathrm{Nat}(F_1\circ G_1,F_2\circ G_2)$?

If $C$ is a monoidal category, there is the canonical map $$ \operatorname{Hom}(A_1,A_2)\times\operatorname{Hom}(B_1,B_2)\to\operatorname{Hom}(A_1\otimes B_1,A_2\otimes B_2) $$ with $(f,g)\mapsto ...
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55 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 ...
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56 views

What is an object with finite length in a tensor category?

I read the definition of a tensor category: A tensor category is a rigid abelian monoidal category in which the object 1 is simple and all objects have finite length. This definition is in "Lectures ...
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54 views

Algebras for monads in Cat and 2-categories

An algebra for a monad $(T, \mu, \eta)$ on a category $\mathbb{C}$ is defined as a morphism $T X \to X$ for some object $X$ such that the obvious diagrams commute. If I look at the monad as a monoid ...
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34 views

Composition of Dual Maps in (rigid) Monoidal Categories

If $X, Y$ are objects in monoidal category $\mathcal{C}$ which have left duals $X^∗, Y^∗$ and $f : X → Y $ is a morphism in $\mathcal{C}$, then the left dual map $f^∗ : Y^∗ → X^∗$ of $f$ is given by: ...
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22 views

Is the category of H-bicomodules within the monoidal category of H-bimodules equivalent to the category of left H-comodules

Fix $\mathbb{k}$ a field. Let $H$ be a $\mathbb{k}$-quasi-bialgebra. Is there an equivalence $ {}_H^H \mathcal{M}_H^H \cong {}^H \mathcal{M}$ where ${}_H^H \mathcal{M}_H^H$ is the category of ...
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36 views

Categorical terminology

For any monoidal category $(\mathscr{C},\otimes,I)$ with objects $A,B,...$: Does the monoidal product $\otimes$ always have the property: $A \otimes A \rightarrow A, B \otimes B \rightarrow B,...$ ? ...
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1answer
40 views

Uniqueness of Dual Objects in Monoidal Categories

I trying to understand the proof of Proposition 2.10.5 from the book: TENSOR CATEGORIES, by P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, See http://www-math.mit.edu/~etingof/egnobookfinal.pdf. ...
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68 views

How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
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1answer
80 views

The algebra of natural transformations of the n-th power tensor functor

Let $k$ be a $0$ characteristic field, $n$ an positive integer and $S_n$ the $n$-th symmetric group. Let's work in the symmetric monoidal category of $k$-vector spaces and linear maps that we denote ...
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38 views

Conjecture concerning involutions in a unitary braided fusion category/Grothendieck ring

Despite the categorical setup, a solution to this question may require no categorical tools (see Conjecture 2). Let $\mathcal C$ be a unitary braided fusion category, $I$ be its set of isomorphism ...
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32 views

Name (reference?) for lax monoidal functors that are 'full, or surjective, on monoids'?

A lax monoidal functor $F$ takes monoids to monoids. Is there a name for a lax monoidal functor that is 'full' or surjective with respect to this property? In other words, the functor $F : \mathcal{C} ...
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What categorical property do these forgetful functors have in common?

Consider the following examples: The forgetful functor $U_1: \operatorname{Vect}_\mathbb{C} \to \operatorname{Vect}_\mathbb{R}$ The forgetful functor $U_2: \operatorname{Diff}^{\text{or}} \to ...
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1answer
330 views

Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
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147 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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48 views

Is Module Category over Monoidal category Monoidal?

let $\mathcal{C}$ be a monoidal category and $\mathcal{M}$ a $\mathcal{C}$-module category. Does $\mathcal{M}$ need to be a monidal category? I know it is true for certain categories, but is it true ...
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64 views

Universality of the Simplex Category. Proving Functoriality of the Map.

Let $B$ be a strict monoidal category, and $\left \langle c,\mu ',\eta ' \right \rangle$ a monoid in $B$. Now suppose we consider the simplex category $\left \langle \triangle ,+,0 \right \rangle$, ...
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Associative Law for a Monoid in a Monoidal Category.

Let $B$ be a monoidal category, and $c$ a monoid in $B$. Powers of $c$ are defined by taking $c^{n}$ to be the $\otimes $-string of length $n$ of $c$ in which the parentheses are all in front. We ...
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36 views

Representation of functions in the simplex category

I am using the following characterization of the simplex category $\Delta$: The objects are finite ordinals and the arrows are weakly monotone functions $f:n\rightarrow m$. $0$ is initial and $1$ is ...
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1answer
88 views

Proving that a category is cartesian closed

Let $Alg(1)$ be a category whose objects are sets with a unary operation, with no axioms. Morphisms of the category are functions of sets which preserve the operation. I need to show that this ...
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43 views

In a closed monoidal category, is $[-,-]$ always a bifunctor?

We say a monoidal category $\mathcal V=(\mathcal V_0,\otimes,I,a,l,r)$ is closed if the endofunctor $-⊗Y$ has a right adjoint $[Y,-]$, called the exponential, for every $Y$. The object $[Y,Z]$ for ...
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Iterated Coproduct in a Monoidal Category; finding the unit of a monoid.

Suppose $B$ is a monoidal category and further that the functors $-\bigotimes a:B\rightarrow B$ and $a\bigotimes -:B\rightarrow B$ preserve coproducts. The we have $\theta :\coprod _{b} a\bigotimes ...
3
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2answers
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Proving associativity in monoidal category: Free Monoid construction.

I am filling in the details of Mac Lane's proof of the following: If monoidal category $B$ has countable coproducts, and if the functors $-\square a$ and $a\square -$ preserve them, then the evident ...
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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 ...