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3
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1answer
40 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 ...
1
vote
1answer
24 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 ...
3
votes
0answers
30 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 ...
2
votes
1answer
47 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 ...
1
vote
1answer
31 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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0answers
49 views

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 ...
1
vote
1answer
36 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 ...
1
vote
0answers
19 views

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 - ...
1
vote
1answer
47 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
votes
1answer
64 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 ...
3
votes
1answer
127 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 ...
2
votes
1answer
49 views

Defining a monoidal category without elements

I am trying to generalize the notion of monoid object internal to a (not necessarily strict) monoidal category, by weakening the associativity and unitarity diagrams (see this nlab entry.) Of course ...
1
vote
1answer
42 views

Rig categories concept which is equivalent to monoid concept in monoidal categories

In monoidal categories, there is a notion of monoid. Is there an "equivalent" concept in rig categories (i.e., categories with two monoidal structures which are related like + and * in a rig)?
13
votes
0answers
126 views

Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
0
votes
0answers
33 views

Tensor structure on the category of algebras for a strong monad

Proposition 1.4 in "Monads on tensor categories" by I. Moerdijk says: "If $S$ is a Hopf monad on a tensor category $\mathcal{C}$, then the category of $S$-algebras is again a tensor category." ...
0
votes
0answers
47 views

Underlying functor of tensor product in a closed and symmetric monoidal category.

I will follow, for terminology and notation, G. M. Kelly, Basic Concepts of Enriched Category Theory. For sake of a self-contained exposition, I will try to write here all the needed concepts. Let ...
0
votes
0answers
54 views

Module and bimodule categories equivalent to a 2-functor

Let $A$ be a tensor category (i.e. monoidal category). A (right) module category of $A$ is a category $M$ with a coherent action $\mu\colon M\otimes A\to M$. Denote $BA$ be the one-object bicategory ...
1
vote
1answer
46 views

Why is a braided left autonomous category also right autonomous?

In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that: Lemma 4.17 ([23, Prop. 7.2]). A braided ...
3
votes
0answers
52 views

Logic in closed symmetric monoidal categories; reference request.

Suppose we want an algebraic theory $T$ to be interpretable in any closed symmetric monoidal category $\mathbf{C}.$ I am thinking in particular of the case where $\mathbf{C}$ is the category of models ...
1
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0answers
33 views

Coherence theorem for symmetric monoidal categories

What's the formal statement for the coherence theorem for symmetric monoidal categories? I've seen there's some notion of permutation around, but I can't get my head around the thing that "all ...
5
votes
1answer
146 views

What is the “opposite” of a forgetful functor?

Consider a category $C$ and a monoid $M$. Consider a functor $F:C\to M$. It maps the objects of $C$ into the only object of $M$. But I don't want it to map every morphism of $C$ into the identity on ...
5
votes
2answers
147 views

Relations between monoids and modules?

What is the relation between monoids and modules? Are they completely different algebraic structures, or is there a kind of inclusion relation like "elements of a module are also elements of a ...
2
votes
2answers
42 views

Distributivity in linear monoidal categories

Let $\mathcal{C}$ be a linear monoidal category, that is a monoidal category (with tensor product $\otimes$) enriched over $\mathbf{Vect}$. Now as far as I can tell the axioms for a linear category ...
7
votes
1answer
104 views

Is dualizablility of an object equivalent to tensoring with that object having a left adjoint?

Let $C$ be a closed symmetric monidal category. There is hence an adjunction $$ -\otimes X\colon C\leftrightarrows C\colon Map(X,-) $$ involving the internal Hom $Map(-,-)$ for every object $X$ of $C$ ...
5
votes
1answer
72 views

Monoidal categories and tensor products

Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts? This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ ...
3
votes
0answers
40 views

Characterization of certain maps in $Hom(A \otimes A^{*}, A \otimes A^{*})$

Let $(M, \otimes, I)$ be a symmetric monoidal category and let $(A, B, \eta, \epsilon)$ be a dual pair in $M$. Consider maps $i_{A}: Hom(A, A) \rightarrow Hom(A \otimes B, A \otimes B)$, $i_{B}: ...
4
votes
1answer
82 views

At a closed monoidal category, how can I derive a morphism $C^A\times C^B\to C^{A+B}$?

Let $A$, $B$ and $C$ be objects of a closed monoidal category which is also bicartesian closed. How can I derive a morphism $C^A\times C^B\to C^{A+B}$? $(-)\times (-)$ denotes the product, $(-)+(-)$ ...
2
votes
0answers
103 views

Every monoidal category is strictly equivalent to a strict monoidal category.

I am reading Joyal and Street's article "braided tensor categories". The following theorem is proved (Theorem 1.2). Let $ C $ be a category. Let $FC$ and $F_sC$ be respectively the free monoidal ...
0
votes
0answers
40 views

Recovering Ordered Monoid Operation from the Order

I have a partially ordered set $(X, \preceq)$ with the following properties: $X$ has a minimum. I'll name it $1$. For every $x \in X$, the principal filter ${\uparrow} x$ is order isomorphic to $X$. ...
3
votes
0answers
70 views

Natural transformation defined by one element

Let $C$ be a self-enriched category, a CCC, and $F : C \to C$ an endofunctor with strength, that is, $F$ comes with a natural transformation $$st_{A,B} : A \times F B \to F (A \times B)$$ such that ...
3
votes
1answer
112 views

Monoid as a one-object category… for monoidal categories

A monoid can be seen as a one-object category. Is there analogous thing for monoids in a monoidal category $(M, \otimes, I)$? Can I form some kind of one-object category from a monoid in $M$?
1
vote
1answer
81 views

Inverse of Braiding in Monoidal Category is Braiding

In Mac Lane's Categories for the Working Mathematician in section $\mathrm{XI}.1$ about symmetric monoidal categories we find that a braided monoidal category $M$ is a monoidal category ...
1
vote
1answer
57 views

What is required for a monoidal category to have products/coproducts?

What is required for a monoidal category to have products/coproducts? If it helps the particular category I am interested in also has zero morphisms.
0
votes
1answer
65 views

$\textbf{C}$-Monoids and products

i have a question about $\textbf{C}$-Monoids. We can make a new category $\textbf{Mon(C)}$ from the category $\textbf{C}$, namely the category of all $\textbf{C}$-monoids. A $\textbf{C}$-monoid is a ...
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vote
0answers
69 views

On the definition of algebra .

Let $\mathscr{C}$ a monoidal category with monoidal product $A\circ B$. Is defined the bicategory of bimodules $Mod(\mathscr{C})$ on $\mathscr{C}$ (see [Gray] p. 46). Its objects are ...
0
votes
1answer
51 views

On the monoidal product of comodules

Let $\mathcal{V}$ a symmetric monoidal category. A commutative comonoid is a triple $(A, \delta, \epsilon)$ whit $\delta: A \to A\otimes A$, $\epsilon: A\to I$ by the dual of monoid axioms. Given two ...
3
votes
2answers
79 views

Understanding associators as natural transformations

Reading Baez and Stay's "Rosetta Stone," and trying to understand the definition of monoidal category on page 12, I read that a monoidal category requires a natural isomorphism called the associator, ...
5
votes
0answers
137 views

Generalization of analytic functors

A functor $F\colon \bf Sets\to Sets$ is said to be analytic if it results from the left Kan extension of a functor $f\colon \mathbf{Bij}(\mathbb N)\to \bf Sets$ (the "species" of the functors $F$) ...
3
votes
2answers
100 views

Proving that tensor distributes over biproduct in an additive monoidal category

I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus ...
5
votes
1answer
153 views

Grothendieck group of a symmetric monoidal category is a lambda ring?

I understand that taking the Grothendieck group of a braided monoidal (abelian) category gives us a commutative ring and that taking that of a symmetric monoidal (abelian) category gives us a ...
3
votes
3answers
221 views

Theory of promonads

I'm led to define a promonad in $\bf D$ as a monoid in the category of endo-profunctors of a category $\bf D$, where the product of two profunctors is their composition as profunctors: $$ F\odot G := ...
4
votes
1answer
58 views

Rooted trees morphisms and categories

If I take the monoidal category freely generated by a single object $A$ and a morphism $f: A \otimes A \to A$, I end up with the monoidal category whose morphisms are forests of binary rooted trees. ...
2
votes
1answer
104 views

Is there a categorical construction of the general linear group?

This question is related to the answer of Qiaochu in this one. Since the object $X=\mathbb{F}_2^2$ generates the category of vector spaces of dimension $2^n$ over $\mathbb{F}_2$, and since we know ...
0
votes
0answers
118 views

Nonsymmetric monoidal product on $[\mathbb N,\mathbf{Sets}]$

Let $\mathbf P$ be the category with objects the natural numbers and $\hom(m,n)=Sym(n)$ if $n=m$, and the empty set otherwise. It is symmetric monoidal wrt the sum of natural numbers, and has $0$ as a ...
3
votes
2answers
235 views

Associativity of Day convolution

I'm trying to follow Day's argument to prove that $[\mathbf C,\mathbf{Sets}]$, where $\bf C$ is symmetric monoidal, is itself symmetric monoidal, but I'm stuck at the very beginning. Is there a way to ...
1
vote
0answers
41 views

Do we need braiding to define a monoid structure on product of monoids?

Let $(\mathcal{C}, \otimes, I)$ be a monoidal, symmetric category. It is well-known that in this case, if $M_{1}, M_{2}$ are monoids, there is a natural monoid structure on $M_{1} \otimes M_{1}$. The ...
4
votes
1answer
118 views

Reasons for coherence for bi/monoidal categories

Here by coherence conditions I mean those axioms imposed on associators and unities that grant that the groupoid generated by such morphisms is a poset, i.e. any two parallels morphisms in this ...
7
votes
1answer
191 views

Why can I choose to work in a strict monoidal category without loss of generality?

Let $\mathcal A$ be a monoidal category. We know that $\mathcal A$ is monoidally equivalent to a strict monoidal category $\mathcal A^{\mathrm{str}}$. In many books/papers it is assumed without loss ...
6
votes
0answers
145 views

Mnemonic device for relationships between Hom and Tensor

Probably this is a stupid question, but nevertheless... Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
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0answers
97 views

Left-modules over a bialgebra form a monoidal category

Let $B = (B, \nabla, \eta, \Delta, \epsilon )$ be a bialgebra over a commutative ring $k$. Let $M$ and $N$ be two left $B$-modules. Then the tensor product $M \otimes_k N$ becomes a left $B$-module ...