The tag has no wiki summary.

learn more… | top users | synonyms

3
votes
1answer
34 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 ...
4
votes
1answer
54 views

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 ...
0
votes
1answer
62 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 ...
2
votes
1answer
29 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 (braided) ...
0
votes
1answer
56 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 ...
1
vote
1answer
16 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 ...
5
votes
0answers
68 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 ...
2
votes
0answers
44 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 ...
3
votes
1answer
49 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 ...
3
votes
1answer
35 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 ...
8
votes
2answers
178 views

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, ...
4
votes
1answer
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 ...
3
votes
0answers
37 views

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 ...
0
votes
1answer
59 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 ...
1
vote
0answers
63 views

Monoidal Categories

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 ...
1
vote
1answer
26 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?
0
votes
1answer
60 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 ...
11
votes
0answers
124 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 ...
2
votes
1answer
90 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 ...
3
votes
1answer
45 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
57 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 ...
4
votes
0answers
53 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
61 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
37 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
0answers
60 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
55 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
23 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
55 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
164 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
133 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
63 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
54 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
150 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
47 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
53 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 ...
1
vote
1answer
50 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 ...
4
votes
0answers
72 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
vote
0answers
38 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 ...
6
votes
1answer
174 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
165 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 ...
3
votes
2answers
49 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
126 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
88 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
48 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
89 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
127 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
42 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
76 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
144 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
91 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 ...