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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159 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 ...
13
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1answer
102 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 ...
13
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1answer
155 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 ...
12
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132 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 ...
9
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2answers
201 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, ...
8
votes
1answer
406 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
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1answer
138 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
222 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
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1answer
179 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 ...
6
votes
0answers
164 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
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2answers
176 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
192 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
235 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 ...
5
votes
1answer
444 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
92 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
120 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
1answer
70 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 ...
5
votes
0answers
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 ...
5
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0answers
184 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
3answers
275 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
2answers
176 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
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 ...
4
votes
1answer
91 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
126 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
61 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
1answer
65 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 ...
4
votes
1answer
65 views

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 ...
4
votes
0answers
69 views

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 ...
4
votes
0answers
58 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 ...
4
votes
0answers
87 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
1answer
159 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
55 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
131 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
1answer
71 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 ...
3
votes
2answers
98 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
312 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
270 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
47 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
135 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
68 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
146 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
1answer
63 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 ...
3
votes
2answers
62 views

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 ...
3
votes
1answer
41 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 ...
3
votes
1answer
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 $\left \langle \triangle ,+,0 \right \rangle$, ...
3
votes
0answers
39 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 ...
3
votes
0answers
51 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
79 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
70 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 ...