Talk:Monoidal category
From Wikipedia, the free encyclopedia
 |
|||||||||||||||||||||||
| Mathematics rating: | Start Class | Mid Priority | Field: Foundations, logic, and set theory | ||||||||||||||||||||
|
|||||||||||||||||||||||
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)
Contents |
[edit] 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
[edit] 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)
[edit] Free strict monoidal category
What is "tensor" in such a category? —Vivacissamamente 23:32, 24 April 2006 (UTC)
[edit] 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
-
[edit] 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.
[edit] Removed example table
In my last edit ([1]) 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)
[edit] 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)
