Talk:Monoidal category

For one sentence - perhaps three mistakes.

- The product in a monoidal category isn't a categorical product (no projection maps to factots, cf. tensor product).

- the associativity is only up to coherence maps.

- The identity object can't be called an identity element.

Charles Matthews 16:47, 1 Nov 2003 (UTC)

This article should explain what a monoidal category is. For example, is the category of all groups, together with the cartesian product, a monoidal category? AxelBoldt 10:43, 18 Aug 2004 (UTC)

Several wrong definitions
0) A collection of correct definitions is in

http://math.ucr.edu/home/baez/qg-winter2001/definitions.pdf

Here I sketch some of the definitions:

1) Strict monoidal category = a category C with a specified object I (unit object) and a bifunctor m: C x C -> C satisfying the axioms of monoid. E.g. m(A,(B,C))=m((A,B),C).

No coherence axioms! That is what strict means.

2) (weak) monoidal category = a category C with a specified object I (unit object) and bifunctor m: C x C -> C satisfying axioms of monoid only up to a canonical natural equivalence (this meant by weak). E.g. m(A,m(B,C)) and m(m(A,B),C) are not the same functors but they are naturally equivalent and this equivalence is part of the structure.

By canonical I mean that the natural equivalences are part of the structure and that however you combine them to obtain new equivalences, you can obtain at most one equivalence between any two functors (this is the content of the coherence axioms).

3) Braided monoidal category = monoidal category with additional structure: a natural equivalence (called braiding): b{A,B}: m(A,B) -> m(B,A) satisfying axioms I am lazy to write here.

Symmetric monoidal category = braided monoidal category with b{A,B} b{B,A} = 1.

In particular, a braided monoidal category is NOT a relaxed version of a monoidal category.

-- Gábor Braun

Attempted cleanup
I have edited the article so that it at least (and at last!) gives a correct definition. It's difficult to know where to stop: one could easily write a whole book about monoidal categories.

There should be some mention of monoidal functors and natural transformations. Actually they probably need an article of their own! The brief allusion to braiding and symmetry should be expanded into a proper definition. Monoidal closed (and compact closed, and star-autonomous) categories deserve a look-in too. The connection with linear logic deserves to be explained.

Then what about premonoidal categories (in the sense of Power), and promonoidal categories? The list goes on...

Still, this is a start.

--Puffinry 23:47, 7 Mar 2005 (UTC)

I've corrected a couple of dubious claims that have crept in here. A one-object 2-category is only a strict monoidal category: the confusion probably results from the fact that some authors (mainly physicists) use the word 2-category to mean bicategory.

I've also removed the definition of lax monoidal category, because it's non-standard. There may well be some authors who use the term in the sense that was given here (citations please?), though I have not seen it myself. But more importantly, there is a tradition of using lax monoidal category to mean a lax algebra for the 2-monad for strict monoidal categories. You see this latter usage in the writings of Tom Leinster and Ross Street, for example.

Puffinry 16:06, 23 April 2006 (UTC)

Free strict monoidal category
What is "tensor" in such a category? &mdash;Vivacissamamente 23:32, 24 April 2006 (UTC)

Tannakian Categories and tensor product!!!
Is a Tannakian category monoidal? Is a Tannakian Category braid and symmetric? I think so but I don't remember...moreover there's something which is not very clear in my mind...for example: why don't we see a third element in tensor product such as $A\otimes_B C$? Is this the same in $Rep(G,k)$ (the category of the representations of an affine $k$-group scheme, which is certainly tannakian)? I've never thought about it...I've always  simply believed, as usual in other categories, that a tensor product is something like $A\otimes_B C$ but I begin to think otherwise!!! So...what a tensor is (as Vivacissimamente told us)? Losc 22:43, 12 August 2006 (UTC)
 * for your third object question: you should think in terms of fibred categories and so there is nothing really new. kalash october 2006

typo in first diagram
In the label on the top-left horizontal arrow, the '\otimes D' should be subscripted. Also the right vertical arrow's label is funky. Thought I'd mention it in case someone still has the diagram's source handy.

Removed example table
In my last edit I was bold and removed an poorly formatted table of diagrams comparing monoid objects in R-Mod and Set. For one thing, the table really belongs at monoid object, not here. Secondly, I don't think the table really adds anything (to this article or that one) since the diagrams are just copies of those shown at monoid object with trivial relabeling. -- Fropuff (talk) 07:10, 22 January 2008 (UTC)

Just checking my understanding...
I haven't read very much of this yet, but: Is the idea just that it's a category in which a reasonable sort of tensor product of objects exists? Michael Hardy (talk) 01:35, 6 May 2008 (UTC)
 * Oh. OK.  I should have read beyond the intro section before asking this.  Well, as Emily Litella would say........... Michael Hardy (talk) 01:37, 6 May 2008 (UTC)

Infinite tensor product?
Is there a definition of infinite tensor product? --VictorPorton (talk) 20:06, 19 September 2015 (UTC)

$$n$$-ary tensor product for finite $$n$$?
I feel that everything in this article can be rewritten with use of $$n$$-ary monoidal product for arbitrary natural number $$n$$ instead of binary tensor product.

If my idea is right, the alternative definitions should be described in details enough.

If it isn't, something should be said in the article on what is wrong with this.

--VictorPorton (talk) 20:27, 19 September 2015 (UTC)

I misunderstand something
$$(A\otimes B)\otimes C$$ and $$A\otimes(B\otimes C)$$ are morphisms, but $$\alpha_{A,B,C}$$ is a natural transformation and thus is defined between functors not between morphisms.

Isn't it a contradiction?

Please explain me my error. --VictorPorton (talk) 21:35, 19 September 2015 (UTC)


 * Nope. The sentence says that the components are $$\alpha_{A,B,C} : (A \otimes B) \otimes C \leftrightarrow A \otimes (B \otimes C)$$. This is shorthand for introducing two functors $$L, R : \mathbf{C} \times \mathbf{C} \times \mathbf{C} \to \mathbf{C}$$ given by $$L \; (A, B, C) = (A \otimes B) \otimes C$$ and $$R \; (A, B, C) = A \otimes (B \otimes C)$$. Then the natural transformation is $$\alpha : L \to R$$, and its components are the $$\alpha_{A,B,C}$$. Wchargin (talk) 22:22, 18 March 2016 (UTC)

The most basic examples that motivate the name are missing.
I think it would be edifying to explain that any monoid can be regarded as a strict monoidal category by taking the discrete category on the elements then making tensor product the multiplication operation. More generally, to see a simple example of a monoidal category that is not strict is to regard a non-free monoid as a quotient of a free monoid. You get additional isomorphisms that correspond to two words being "equal" in the quotient. — Preceding unsigned comment added by Ralth (talk • contribs) 23:17, 27 April 2017 (UTC)

Spam in reference
A user is repeatedly putting links to a document titled "Maclane pentagon is some recursive square" with the apparent aim of promoting the author of that work, which is spam according to wikipedia guidelines. This is also happenitng to Coherence condition and Coherence theorem. Antonfire (talk) 07:26, 8 September 2017 (UTC)

Should we remove _tensor categories_
Often on wikipedia we can read "tensor category", whereas the authors really mean "monoidal". There is not really a consensus, but usually people actually mean something with way more structure. For example abelian monoidal with exact tensor functor and stuff like that. So I think we should refrain - across all of Wikipedia - from using "tensor category", and instead use the un-ambiguous "monoidal category". 134.100.222.237 (talk) 09:38, 21 June 2018 (UTC)

Some nonsense in the third paragraph in the introduction
In the introduction, there is a vague, hand-wavy paragraph describing monoidal categories from the perspective of something completely random. Specifically the paragraph that begins as "A rather different application, of which monoidal categories can be considered an abstraction, is that of a system of data types..." I think it's fair to say that this is (1) irrelevant to the general idea of a monoidal category and (2) both confusing and very unhelpful for readers who are visiting this page.

Unless anyone has any objects. I would like to remove it, or alternatively push it down away from the introduction. Lltrujello (talk) 01:56, 28 November 2020 (UTC)

"Algebra/set analogy" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Algebra/set analogy and has thus listed it for discussion. This discussion will occur at until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Steel1943 (talk) 20:02, 6 January 2023 (UTC)