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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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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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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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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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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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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72 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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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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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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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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102 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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40 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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59 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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67 views

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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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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75 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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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 ...
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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 ...
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86 views

Natural transformations and the definition of Monoidal lax functors

The definition of a lax monoidal functor requires the existence of a natural transformation, $\phi$ http://en.wikipedia.org/wiki/Monoidal_functor. A natural transformation relates at least 2 ...
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Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
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69 views

every category is equivalent to its universal cover

I am just curious how could we show that every category is equivalent to its universal cover. To me, it is not obvious how could we assign to each an object in a category $\mathcal{C}$ to a family of ...
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46 views

Coherence result for (braided) monoidal functors

Is there any coherence result for (braided) monoidal functors? (like Mac Lane's coherence theorem for monoidal categories) What I have in mind is a theorem like the following: Let $F$ be a ...
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83 views

About the definition of Day's convolution

I'm struggling with the definition of Day's convolution. Given a monoidal category $(\mathcal C,\otimes, I)$, there is a way to define a monoidal product on the category of presheaves ...
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1answer
18 views

Does “modular category” make sense without saying “abelian” or “linear”?

I know the term "modular category" only from representations of quantum groups, TQFTs and fusion (finitely semisimple linear) categories. There, a modular category is a ribbon fusion category where a ...
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69 views

Tensor of tensored categories

Given two $V$-categories $C$ and $D$ tensored over a symmetric monoidal category $V$, could I form the "tensor" of $C$ and $D$? More precisely, is there a $V$-category $T(C,D)$ such that $V$-functors ...
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50 views

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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65 views

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ as a coproduct?

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ is defined grading-wise by $$(A\Rightarrow B)_n=\prod_{i\in \mathbb Z} \text{Hom}_R(A_i, B_{i+n})$$ Intuitively, I would have defined the ...
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36 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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Tensor products from internal hom?

Monoidal categories come with tensor products, and sometimes, these categories are biclosed, i.e each restriction of the tensor bifunctor has a right adjoint. If the category happens to be symmetric, ...
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32 views

Are units in rigid (autonomous) categories some sort of natural transformation?

In a rigid category $\mathcal{C}$, let us choose left and right duals and left and right (co)units for every object. This gives us, for example, a dualisation functor $-^*:\mathcal{C} \to ...
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What are coquantales?

A quantale can be defined as a monoid in the monoidal category $\mathbf{Sup}$ of complete join semilattices, equipped with tensor product. What are examples of non-diagonal comonoids in ...
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61 views

References about “algebras over monoids”

Please, could someone point me any reference (with a bit of details) about "algebras over monoids" (in the sense of Schwede & Shipley, Algebras and modules in monoidal model categories)? Thank you ...
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88 views

Showing $\lambda_1=\rho_1$ in monoidal category

For a monoidal category $\mathcal{C}$ with $\alpha_{a,b,c}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c$, $\rho_a : a \otimes 1 \rightarrow a$, and $\lambda_a: 1 \otimes a ...
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1answer
30 views

Is there a special name or any research on Cartesian compact closed categories?

As per the title. I can't find anything about the combination of the two, and such categories interest me. Does anyone know of any such categories?
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62 views

Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$

I'm looking for examples of monoidal categories $\mathbf{C}$ such that one of the following two statements holds. For all objects $X$ and $Y$ of $\mathbf{C},$ $\mathrm{Aut}(X \otimes Y) \cong ...
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How to name these “ideals”?

Background. Let $\mathcal{C}$ be a symmetric monoidal category with unit $\mathbf{1}$. A subobject of $\mathbf{1}$ is just a monomorphism $I \to \mathbf{1}$. We may also call this an ideal of ...
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1answer
100 views

Tannaka reconstruction: reference request

What is a classical and perhaps even original reference for the following result, often called Tannaka reconstruction? Let $G$ be a group and $R$ be a commutative ring in which $0,1$ are the only ...
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58 views

Uniqueness of Duals in a Monoidal Category

Given a monoidal category ${\cal C}$, and $X \in {\cal C}$, we define a left dual of $X$ to be an object $X^*$ such that there exist morphisms $\epsilon:X^* \otimes X \to I$, and $\eta:I \to X \otimes ...
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1answer
86 views

Mac Lane‘s proof of coherence theorem for symmetric monoidal categories.

In [CWM, Ch. XI, §1], Mac Lane prove the coherence theorem for symmetric monoidal categories by assuming the strictness. Thus we have a $n-$ary tensor functor $T$, and the theorem states that any two ...
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69 views

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 ...
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65 views

Connections of Finite groups and quantum groups

I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I ...
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43 views

Monoidal categories and Generators

Let $\mathcal{C}$ be a Cocomplete Cowellpowered Monoidal category. Does $\mathcal{C}$ need to have a generator? I think it does not, but it seems hard to get a counter example.
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Taking the quotient of a free monoidal category modulo a relation

I have recently been advised, in a particular example which is described below, that taking quotients of a category can be complicated, but I don't see where exactly lies the difficulty. More ...
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1answer
72 views

Coherence in braided monoidal categories

Let ($\mathcal{C}$,c) be a braided monoidal (tensor) category. Then c is compatible with the morphisms l,r associated with the unit object 1 of $\mathcal{C}$, in the sense that: $l_X \circ ...
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Symmetries and internal hom in monoidal categories

I need some help in trying to define an object. Suppose we have a biclosed monoidal category (possibly non-symmetric) and two natural isomorphisms: $B \otimes - \simeq - \otimes B, \ \ C \otimes - ...
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1answer
67 views

Why are duals in a rigid/autonomous category unique up to unique isomorphism?

I'm having trouble understanding the following statement: "In a rigid category, duals are unique up to unique isomorphism." It seems to me that this isomorphism is not unique. Let me try to give a ...
2
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1answer
209 views

Monoidal product is coproduct in category of commutative monoids

If $V$ is a symmetric monoidal category, the category $\text{CMon}(V)$ of commutative monoids in $V$ has binary coproducts given by $\otimes$, the monoidal product of $V$. See for example Johnstone’s ...
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137 views

Natural Isomorphism: how can $A \otimes B \simeq B \otimes A $ and yet $A \otimes B \neq B \otimes A $

I am reading Braided Monoidal Categories by Joyal and Street. They say cateogories with tensor product arise naturally such as the category of Abelian Groups and that of Banach Spaces. Is there any ...