Talk:Category theory

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[edit] A question on functors

Must a functor be one-to-one? Can it assign the same object in D to many objects in C, or many objects in D to a single object in C? I can't see that this is ruled out by the requirements on a functor, but maybe I'm just not being smart enough.
-- Stuart Presnell 28 Mar 2003

A functor can be many-to-one but not one-to-many, so much like a function really. It isn't actually a function because functions are defined on sets and categories are generally 'bigger' than sets. Pcb21 14:06 31 May 2003 (UTC)

Actually, it isn't actually a function because it is required to preserve structure (objects, functions, and their pattern of connections).

[edit] Category theory is not only "a language" !!

Sorry, if my English is too bad you can...

- There are books much more readable than "categories for the working mathematician". I propose, like does "super" John Báez (UCR), this book: "William Lawvere and Steve Schanuel, Conceptual Mathematics: A First Introduction to Categories, Cambridge U. Press, Cambridge, 1997."

- The theory is not only a major challenge for the "serious mathematicians" (a serious mathematician is also interested on "foundations", he not only is working in concrete "mathematical" problems), the theory is also a great event in philosophy, and it is a beautiful and powerful tool for "thinking". The article in Wikipedia is too cold and "mathematical". Mathematics depends on a decision, and then... deduction and more deduction. But today mathematics and science in general, have -within it- a medium that can "to bring" "consciousness" and have a marvellous "compass" neccesary for their interminable work. Also I hope that the theory can enormously help in "learning/teaching" about anything.

- I think that phrases like "abstract-nonsense" are signs of "reactionary will". All maths are "abstract".

Thank you very much. --ivian 09:29, 20 May 2004 (UTC) ivan.domingo(arroba)gmail.com

Well, no. You could say abstract nonsense is rather a dated phrase. But mathematics contains an entire range of abstraction, from a hands-on tiling question to topos theory.

There is another, more technical point, about whether category theory is really more like Mac Lane or Grothendieck's visions; or more combinatorial stuff (cf. De Bruijn notation). Again it is both, really.

Charles Matthews 09:37, 20 May 2004 (UTC)

[edit] Structure of the article

Something else: I think it would be useful for the structure of the page to export some of the topics (the numerous examples are great but the presentation becomes not very concise). Especially "Natural transformations" could go to Natural transformation and there should be an extra page for "Equivalence of Categories", since there is really much more to say about this. The page would be clearer by just pointing to these topics (and giving very short explanations).

Comments welcome --- Markus 21 Nov 2003

OK, no comments, but I did it anyway. I did not destroy any information but added some (especially on equivalence of categories). I think it is a much better structure now, also in the light of further extensions. Still the most basic definitions remain in category theory, but important further topics are now more easily recognized (not just by reading the examples...). I also included the suggested literature. Still comments are welcome.

--- Markus 25 Nov 2003

[edit] Mor(-,-) vs. Hom(-,-)

The set of morphisms is first introduced as Mor(-,-) and later called Hom(-,-). I consider Hom to be the better notation (its kind of standard, isn't it?). Or do we want to change the name of this functor depending on whether the considered set are actually homomorphisms in an algebraic sense? The later would be quite strange in my oppinion, since in the abstract setting of category theory, one usually does not emphasize the internal structure of the objects/morphisms.

-- Markus 25 Nov 2003

I've seen both, but have seen Hom(-,-) much more often than Mor(-,-). I suppose it's a matter of taste, my personal opinion would be that Hom(-,-) be given as the standard notation, with the mention that Mor(-,-) is an acceptable alternative notation. Revolver 01:39, 17 Mar 2004 (UTC)

In general a category C consists of objects and morphisms between objects. Morphisms are called homomorphisms if the category has more structure. If I remember it right, then morphisms of additive categories are called homomorphisms.

(MacLane 1971) talks about hom-sets wrt. Set (p60), so the nearest we have to a canonical reference doesn't recognise the distinction. The filters "site:www.mta.ca inurl:catlist" gets google to search postings to the categories mailing list from June 1994 to December 1999: the additional term "mor" gets 0 hits, while "hom" gets 42 hits (this of course doesn't get all postings, obnly those linked to from elsewhere, but the message is the same): the the Mor(-,-) usage is definitely an oddity. I propose we switch to Hom(-,-). ---- Charles Stewart 06:12, 22 Sep 2004 (UTC)

I'd vote for switching. Though I have seen Mor(-,-) used occasionally in texts, it's very rare. And I've never heard the phrases "mor-set" or "mor-functor". This will involve a lot of page edits though. I'm not volunteering. -- Fropuff 14:09, 2004 Sep 22 (UTC)

Actually I've seen C(A,B) a lot for the set of morphisms in C from A to B. I think it's nice, but I don't know how common it really is. Bgohla 23:20, 2005 May 3 (UTC)

[edit] Dual versus opposite

Following a question I asked in Talk:Equivalence_of_categories, I'd like to suggest we move from talking of the dual of a category to the opposite of a category: while the usage dual category is recognised, opposite category generally appears to be regarded as more correct (see, eg. p.31, MacLane 1971), and is consistent with the notation we use. ---- Charles Stewart 05:41, 22 Sep 2004 (UTC)

[edit] Module (category theory)

Hi there! Does the concept of Module (category theory) make any sense to you? If so, would anybody write an article about this? This article seems to be requested on Wikipedia:Requested articles/mathematics. I suspect the person requesting this article confused something, but I could be wrong. Thanks. Oleg Alexandrov 00:16, 29 Dec 2004 (UTC)

[edit] Non-technical explanation

I wouldn't expect non-technical readers to really understand category theory after reading this article, but from the start of the article until when one gives up reading because one is totally lost, *some* hint of what the subject of the article actually is should be obtained.

I think it would help if the Examples sections were moved up. Also, if there is a a real-world analogy that the Person on the Street can say, "Ah, categories are kind of like that." that would be very helpful.

The current introduction gives the impression that category theory is controversial among mathematicians and may be false or non-rigorous. Since the rest of the article doesn't bear that out, perhaps something like the following should be added to the end of the intro:

The use of this term does not mean that mathematicians consider category theory to be fuzzy or non-rigorous, merely that it is too complex to follow the details in casual conversation.

-- Beland 07:24, 9 Jan 2005 (UTC)

I added an analogy as you suggested to the background section and also the clarification to the intro. I think that should help.

--Carl 01:52, 22 Jan 2005 (UTC)

[edit] Locally small categories.

The collection of morphisms from object to another does not form a set by definition. When it does form then category is called locally small as stated in book by Michael Barr and Charles Wells .

[edit] n-categories

Anyone brave enough to start an article on n-categories?

Try n = 2 first. Charles Matthews 17:16, 24 Mar 2005 (UTC)

[edit] Current state of the article

It needs a severe edit, no? Charles Matthews 18:50, 18 Apr 2005 (UTC)

I would say it could do with some more coherency. Much of the material that used to be in here has now been moved to its own page (e.g. category (mathematics), morphism, functor, natural transformation). The article now reads in a rather choppy fashion. In my opinion, this article should give an overview of the history and motivation for category theory with brief blurbs on the main concepts and links to the appropriate articles. -- Fropuff 19:29, 2005 Apr 18 (UTC)
Category theory is half-jokingly known as "generalized abstract nonsense".
It seems to me that leaving this statement stand without explanation could pose problems. A lot of math students, when first hearing about categories, only hear phrases such as this and hand-waving used in proofs, and many of them DO have the impression that it really is "all just a bunch of abstract nonsense". Revolver 21:35, 18 Apr 2005 (UTC)
It's a fair warning, then. We should get deeper into the debate in the article, actually: what Mac Lane believed and so on. Charles Matthews 07:00, 19 Apr 2005 (UTC)

[edit] Math markup

The article uses the Latin letter 'o' for function composition (g o f) which looks kind of dumb. The proper Unicode symbol is U+2218, or &#x2218; in HTML (g ∘ f), as anyone can easily find out. Is there any reason not to use it? The function composition article uses <small>o</small>, by the way (g o f).
—Herbee 18:48, 2005 Apr 20 (UTC)

The proper symbol (U+2218) doesn't display correctly on some browsers, most notably Internet Explorer. On IE it displays as a small box. -- Fropuff 19:14, 2005 Apr 20 (UTC)

[edit] Mistake in the definition!?

Well, for any tow objects A,B in a category C, Hom_C(A,B) should be a set not a class, at least this is the normal definition. I'v not see a definition where Hom_C(A,B) is allowed to a class. But maybe it's me? The collection of all C-morphisms (the union of all the sets Hom_C, is that what is meant by hom(C)?) is a class. If we allow Hom_C(A,B) to be a class, then the collection of all C-morphisms will be a conglomerate (collection of classes). So, i think, the second line of the definition should go something like. For any to objects A,B a set Hom_C(A,B)....

Some authors allow a more general definition where Hom(A,B) is indeed allowed to be a class. These authors refer to categories where Hom(A,B) is always a set, as locally small categories. -- Fropuff 20:34, 7 March 2006 (UTC)

Ah i was not aware of that. But, if indeed Hom(A,B) is allowed to be large classes, how then can Hom(C) be a class? Should it not be a conglomerate? Don't we get a kind of Russels paradox if we claim Hom(C) to be a class. For instance let U be the collection of all classes, and assume U to be a class. Then consider the class A costing of all classes x \in U for which x \in x. Then A\in A iff A \not\in A. MaVincent 06:35, 8 March 2006 (UTC)

If you read the definition carefully, you'll see that's not what's going on here. One starts with a class Hom(C) that contains all morphisms in C. The class Hom(A,B) is then defined as the subclass of those morphisms with source A and target B. That is to say, Hom(C) is not a class containing other classes but rather a (disjoint) union of classes. I'm not a set-theorist so I'm a little fuzzy how exactly how classes work but think my explanation is essentially correct. Someone please correct me if I'm wrong. -- Fropuff 07:01, 8 March 2006 (UTC)

[edit] object

Under the header "Categories, objects, and morphisms", morphisms is the only one with its own header. It would make much more sense if each of those words had its own definition so that information could be more easily understood. Fresheneesz 01:44, 17 April 2006 (UTC)

[edit] Category of trees

Reposting from User:Lethe/list of categories:

Anyone know if there's been a category defined for trees/binary trees? I note that the p-adic numbers, as well as the real numbers when represented as strings of integers, are actually trees (viz yea olde 0.999..=1.000... debate). I also note that grassmanians (and thus supernumbers, supermanifolds and other bits of supersymmetry) can be represented as binary trees (although not uniquely/naturally). The various cantor sets are also binary trees, as is alpeh_one = powerset of aleph_zero. In a more hand-waving way, it also resembles a recursive application of a Subobject classifier (In the cae of a cantor set, "is it on the left or the right?"). Topologically, this seems like a special case of lattice (order). linas 23:47, 20 July 2006 (UTC)

[edit] Joke in the introduction

I think the joke really should not be in the introduction. As someone pointed out above, it gives the impression that Catgory theory is somehow considered a joke to mathematicians (as if its numerology or something.) I'm not qualified to say if its a fruitful line of questioning as I'm still struggling with set-theory, but I don't think its apporpriate to give such an impression to an intrested laymen especially when there are apparently many mathematicians who take category theory seriously. Or atleast some. Even if it was a fringe theory it still would not be appropriate to have such a joke in the intro. As it does seem pretty clear to me to be innapropriate I'm going to take the liberty of removing it. If someone feels the joke is worth mentioning, please mention it as an aside at the bottom of the article (and source the joke if possible. I think Douglas Hoffstadter may have started that joke or made it popular; I think I remember reading it in one of his sci-am Metamagical Themas columns, so try to source it if you put it back.). Brentt 12:23, 16 August 2006 (UTC)

The 'joke' goes back long before Hofstadter. Please don't just remove it. If you want to qualify the intro, go ahead. Charles Matthews 12:26, 16 August 2006 (UTC)
I agree with Brentt, the "joke" should not be in the intro, especially the part mentioning something about non-sense absolutely irrelevant. Please remove. Tamokk 20:45, 3 September 2006 (UTC)

P.S. I remember reading in "Algebra" by S. Lang, that the term is due to Steenrod. So he refered to the homological algebra done in categories (himself being one of the creators of this theory), rather than the category theory itself, which by then was in a very early stage of development, and was seen by mathematicians as no more then a mere language. Latter the term has been not always neutral, used in different context and with different attitudes (sometimes unsypathetic). Tamokk 20:45, 3 September 2006 (UTC)

[edit] Article rating B+

This is a good article for a technical subject. It may be slightly more technical than is necessary in the beginning parts; the intro (the background section) at least should be readable by an undergrad.

  • The second para of the lead is surprising to me; I know the phrase "abstract nonsense" but it is a stretch to say that "nonsense" refers to commutativity of diagrams. If this is what the original statement meant, I think a citation would be useful to back it up.
  • The historical notes section needs several inline citations, especially for the "it has been claimed" part. Well-known and accepted facts don't need inline cites in my opinion, but direct quotes or attributions of opinion do. The fourth para of that section also needs citations, since the fact that one book was received better than another is not a well-known mathematical fact.
  • The section on Categories, Objects, and Morphisms is very terse; one or two introductory sentences would make it more readable.
  • Here are a few questions that a naive reader might ask. What are contemporary trends in category theory? How is it related to computer science?

CMummert 14:52, 25 October 2006 (UTC)

I certainly agree with the rating the writing in the introduction does a good job of explaining the concept with a minimum of jargon. I wonder if there is any way it could be made visually more appealing? --Salix alba (talk) 15:24, 25 October 2006 (UTC)
The thing about diagram chasing strikes me as possibly original research. I think it need sourcing, at least. And I'll add more about diagrams. Charles Matthews 15:45, 25 October 2006 (UTC)
I think that a section of commutative diagrams in this article, maybe in summary style, would be nice, because they are the picture that most people get when they hear the phrase category theory. The article on commutative diagrams is very short right now, and could use attention. CMummert 15:55, 25 October 2006 (UTC)

I agree that this is a pretty good intro. I have tried to achieve an understanding of category theory a number of times, and this is the article that has brought me closest. --P3d0 02:41, 5 November 2007 (UTC)

I'm working on wikiversity learning project Introduction to Category Theory, maybe it helps you understand, if you're not scared of math... Tlepp 19:10, 5 November 2007 (UTC)

[edit] Portal:Category theory

I added {{Template:Portal}}. The portal is still a "stub", any help is welcome. I added {{Template:Portal|Category theory}} at the pages of the main subjects of category theory too. Cenarium (talk) 17:25, 14 February 2008 (UTC)