Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [monoidal-categories]

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: http://en.m.wikipedia.org/wiki/Monoidal_category)

24
votes
2answers
2k 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 ...
21
votes
0answers
271 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 ...
16
votes
1answer
250 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 $\mathbf{...
14
votes
1answer
233 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
1answer
326 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 $C$ ...
9
votes
1answer
165 views

Categorification of algebra structures

This might be a bit of a soft question. Take a $\mathbb{C}$-linear category. Form the complex vector space spanned by its objecs modulo exact sequences. This construction is, as far as I know, the ...
9
votes
2answers
611 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, ...
9
votes
1answer
414 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 ...
8
votes
3answers
2k views

Day convolution intuition

In the nLab, Day convolution is introduced as a generalisation of convolution of complex-valued functions, but I'm wondering how exactly to understand this. I can (just about) parse the definitions, ...
8
votes
1answer
655 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 \...
8
votes
1answer
236 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 ...
7
votes
3answers
556 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 := \...
7
votes
1answer
134 views

Monoid in the category of endofunctors and Monoid as a category with one object

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...
7
votes
2answers
286 views

Why are invertible objects reflexive in a tensor category?

I am reading Deligne and Milne's notes on Tannakian categories. I'm only just getting used to the idea of an abstract tensor category, and I've encountered a very believable statement that I'm ...
7
votes
1answer
476 views

Details in applying the Barr-Beck monadicity theorem to Tannakian reconstruction

The Barr-Beck monadicity theorem gives necessary and sufficient conditions for a category $\mathcal{C}$ to be equivalent to a category of (co)algebras over a (co)monad. A functor $F:\mathcal{C}\to\...
7
votes
1answer
298 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 ...
7
votes
1answer
109 views

Rings that cannot be representations rings

Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo ...
7
votes
0answers
459 views

What's there to do in category theory?

I'm sure anyone who's heard of categories has also heard the classical "Well obviously there aren't any real theorems in category theory, it's much too general", or something in the likes of it. Now ...
7
votes
0answers
104 views

What's a double category with one object?

Categories with one object are equivalent to monoids. $2$-categories with one object are equivalent to monoidal categories. Therefore, I am wondering whether double categories with one object are ...
7
votes
0answers
219 views

Strongly unbiased symmetric monoidal category

Let $\mathcal{C}$ be a category. Define a strongly unbiased symmetric monoidal structure on $\mathcal{C}$ to be a rule which associates to every finite set $I$ a functor $\mathcal{C}^I \to \mathcal{C}$...
7
votes
0answers
88 views

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 \...
7
votes
0answers
234 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 ...
6
votes
3answers
324 views

Examples of asymmetrically braided monoid

From nCatlab https://ncatlab.org/nlab/show/braiding : Any braided monoidal category has a natural isomorphism $$B_{x,y} \;\colon\; x \otimes y \to y \otimes x $$ called the braiding. ...
6
votes
2answers
210 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 monoid"?...
6
votes
2answers
375 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 ...
6
votes
1answer
140 views

Does $f\otimes \operatorname{Id} = \operatorname{Id}$ imply $f= \operatorname{Id}$?

Let $R$ be a commutative ring, and $X$ an $R$-module. If an $R$-endomorphism of $X$ satisfies $f\otimes \operatorname{Id}_X = \operatorname{Id}_{X\otimes X}$, is it true that $f=\operatorname{Id}_X$ ? ...
6
votes
1answer
400 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 $\lambda$...
6
votes
1answer
290 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 $_RMod$, $Grp$ $CRings$ ...
6
votes
1answer
287 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
1answer
195 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 ...
6
votes
1answer
290 views

Tensor products and morphisms

Let $C$ be semisimple category with simple objects $X_1, \dots, X_r$. Suppose we have a fusion relation $X_i\otimes X_j =\bigoplus_{l=1}^r N_{ij}^l X_l$. Let $m\in \mathbb{N}$ and let $g:mX_j \to ...
6
votes
0answers
110 views

Actions of groups on categories

Let $\Gamma$ be a group and denote by $\underline{\Gamma}$ the tensor category whose objects are simply the group elements, hom-sets only contain identities and the tensor product is given by the ...
6
votes
0answers
90 views

Reconstruction of monoidal categories

Both this post on mathoverflow and this wikipedia page claim that you can reconstruct a monoidal category from its Grothendieck ring and $6j$-symbols (or equivalently the associator). Bruce Westbury ...
6
votes
0answers
91 views

What natural monoidal structure and braiding exists on the category of modules of the convolution algebra of an action groupoid?

Let $S$ be a set with an action $\triangleright$ of a finite group $G$. The action groupoid $S // G$ has as objects the set $S$, and the morphisms from $s_1$ to $s_2$ are just the $g \in G$ that ...
5
votes
1answer
929 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
336 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
175 views

Pentagon diagram of Monoidal Categories

In Wikipedia we can find the following "Pentagon Diagram" for Monoidal Categories. What I don't understand is why that diagram is constructed in that way. Why do we start from $((A \otimes B) \...
5
votes
1answer
165 views

Temperley-Lieb Category

I'm having trouble understanding how the following are exactly related to one another: $\bullet$ The Temperley-Lieb Category TL $\bullet$ The idempotent completion (Karoubi Envelope) of TL, Kar(TL) ...
5
votes
1answer
115 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, $(-)+(-)$ ...
5
votes
2answers
135 views

In what sense right dual and braiding structure respect the tensor product structure in a monoidal category?

Throughout let $(\mathscr{C}, \otimes, \mathbf{1})$ be a monoidal category (I suppressed unitors and associators for simplicity). The usual definition a rigid monoidal category is done in two steps: ...
5
votes
1answer
127 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 ...
5
votes
1answer
48 views

Monoidal structures on small categories and how to find them

Can someone confirm that the category $(\bullet \to \bullet)$, the "walking arrow", has a monoidal structure, but $(\bullet \rightrightarrows \bullet)$ does not? Is there a general method how to ...
5
votes
0answers
50 views

Intuition of factorization systems' role in the construction of free algebras

Kelly's article uses factorization systems for constructing free monoids. What's the intuitive reading of factorization systems in this case? Also, for example, Barr in this article cites a result by ...
5
votes
0answers
88 views

On group graded algebras and Brauer groups

I was reading the paper "Algebras graded by groups" by Knus. I want to test and further my understanding of the paper by asking several questions. Since the paper is not readily available I will ...
5
votes
0answers
97 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
votes
0answers
67 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}: Hom(...
5
votes
0answers
323 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
658 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 ...
4
votes
2answers
112 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 ...
4
votes
2answers
129 views

Categories without a tensor product

What is an example of a category that is useful in geometry and which does not have a tensor product, i.e. a category which we do not know how to turn into a monoidal category? (I am excluding ...