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13
votes
0answers
135 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 ...
12
votes
0answers
130 views

What is the decategorification of a triangulated category?

The decategorification of an essentially small category $\mathcal C$ is the set $\lvert\mathcal C\rvert$ of isomorphism classes of $\mathcal C$. If $\mathcal C$ carries additional structure, then so ...
11
votes
0answers
107 views

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 ...
8
votes
1answer
364 views

Coherence for symmetric monoidal categories

Let $\mathcal{C}$ be a monoidal category. By Mac Lane's coherence theorem for monoidal categories, there are strong monoidal functors $F : \mathcal{C} \to \mathcal{C}_s$ and $G : \mathcal{C}_s \to ...
7
votes
1answer
112 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 $C$ ...
7
votes
1answer
200 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
152 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 ...
5
votes
2answers
157 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 ...
5
votes
1answer
156 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 ...
5
votes
1answer
384 views

Drinfeld Center

Let $\mathscr{C}$ be a strict monoidal category. I will denote the product of $\mathscr{C}$ by $\otimes$. The Drinfeld center $\mathscr{Z(C)}$ of $\mathscr{C}$ is the category with object $(X,\phi)$ ...
5
votes
1answer
76 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 $_RMod$, $Grp$ $CRings$ ...
5
votes
1answer
150 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
1answer
100 views

Coherence Conditions and Strict Monoidal Equivalences

Consider the following: two monoidal categories $({\cal C},\otimes)$, and $({\cal D},\odot)$, and a functor $F:{\cal C} \to {\cal D}$, that gives an equivalence (of ordinary categories) between ${\cal ...
5
votes
0answers
150 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 $F$) ...
4
votes
1answer
217 views

Reference request: Deligne's reconstruction theorem

I've heard this result referenced a few times on MO now. It is supposed to be a theorem of Deligne that gives some natural conditions under which an (abelian?) tensor category $C$ is the category of ...
4
votes
2answers
156 views

“Change-of-base” between enriched categories

I would like to prove that a monoidal functor $$\Phi\colon \mathbf{V}\to \mathbf{V'}$$ induces a functor $$\Phi^\#\colon \mathbf{V}\text{-Cat}\to \mathbf{V'}\text{-Cat}$$ and in particular I ...
4
votes
1answer
84 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, $(-)+(-)$ ...
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 ...
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. ...
4
votes
0answers
33 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 ...
3
votes
1answer
122 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 $M$?
3
votes
2answers
45 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 ...
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 ...
3
votes
3answers
234 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 := ...
3
votes
2answers
86 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, ...
3
votes
2answers
250 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 ...
3
votes
1answer
243 views

What is the categorical perspective on representations of topological groups?

One categorical definition of a group $G$ is that it is a category $C$ with a single object $X$ such that every morphism in the set $C(X,X)$ is invertible, i.e. such that $C(X,X)$ is precisely the ...
3
votes
1answer
41 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 ...
3
votes
1answer
131 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 ...
3
votes
1answer
58 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 ...
3
votes
1answer
145 views

May left and right unitors be equal in a monoidal category?

Monoidal category. $\lambda_I = \rho_I : I\otimes I\to I$? If this equality can not be proved, in what categories it is false?
3
votes
0answers
53 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 ...
3
votes
0answers
42 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}: ...
3
votes
0answers
73 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
0answers
66 views

Cartesian monoidal functors

Let $\mathcal{C}$ and $\mathcal{D}$ be categories with finite products, and consider them as monoidal categories in the obvious way. Every functor $\mathcal{C} \to \mathcal{D}$ can be canonically ...
2
votes
2answers
224 views

$(-1)\otimes (-1) \cong I$

Is there a monoidal category $\mathcal C$ whose unit object is $I$ (i.e. $I\otimes A\cong A\cong A\otimes I$ for all $A\in \text{Ob}_\mathcal C$), with an object "$-1$" such that $$ ...
2
votes
1answer
85 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 ...
2
votes
1answer
54 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 ...
2
votes
1answer
99 views

Is the inverse to a monoidal equivalence also monoidal?

Let ${\cal C,D}$ be two categories, and let $$ F:{\cal C} \to {\cal D}, ~~~~~~~~~~~~~~~~~ G:{\cal D} \to {\cal C}, $$ be an equivalence of categories. Let us now further assume that ${\cal C}$ can ...
2
votes
1answer
126 views

Endomorphisms in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category generated by one element $X$ such that $End(X)=G$ where $G$ is a finite group. Is it true that, for any object $A \in \mathcal{C}$, $End(A)$ is ...
2
votes
1answer
105 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 ...
2
votes
1answer
84 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 ...
2
votes
0answers
108 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 ...
2
votes
0answers
87 views

Thompson's group F and monoidal categories

Fiore and Leinster have proved that if $\mathcal{A}$ is a free monoidal category generated by one object $A$ such that there exists an isomorphism $\alpha: A \otimes A \to A$, then for every object $X ...
1
vote
2answers
189 views

The abstract definition of commutative monoids

In trying to begin to understand the idea of a $k$-tuply monoidal $n$-category, I'm already a bit stuck on the idea (Baez, nLab) that a commutative monoid can be defined as a monoid object in the ...
1
vote
1answer
32 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.
1
vote
1answer
74 views

Morphisms in a symmetric monoidal closed category.

Let $\mathcal C$ be a symmetric monoidal closed category. This means that every functor $- \otimes B$ has a right adjoint $[B, -]$. Let $I$ be the unit and let $\rho \colon - \otimes I \to 1_{\mathcal ...
1
vote
1answer
20 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?
1
vote
1answer
45 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)?
1
vote
1answer
42 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 ...