Talk:Category theory

English, please?

An introductory section in English that someone with 'only' beginning college math might understand is desireable. J. Noel Chiappa 07:24, 18 May 2008 (CDT)

I've drafted something. Does it help any? Criticize away, ... Peter Lyall Easthope 14:55, 18 May 2008 (CDT)

I'm having a bit of struggle seeing the common thread among the examples in the intro - and between them and the examples at the end of the article. Perhaps you could explain the concept in words, at slightly more length than "two mathematical concepts .. the object and the map or morphism"? Having done that, having some examples following that text might then be more illuminating. J. Noel Chiappa 15:24, 18 May 2008 (CDT)

Noel, is this any better? Now an introductory essay rather than paragraph.

Languages such as English have nouns and verbs. A noun identifies an object while a verb identifies an action or process. Thus the sentence "Please lift the tray." conjures an image of a tray on a table, a person who can lift it and the tray in its elevated position.

In a pocket calculator, a datum is a number or pair of numbers. The calculator has a selection of operations which can be performed. Given the number 5, pressing the "square" key produces the number 25.

High school mathematics introduces the concepts of set and function. Given the function

f(a) = a²

we know that the solution set for

f(a) = 25

is

{-5, 5}.

The mathematical abstraction drawn from these examples is based on two concepts: objects and the things which act on objects. In category theory, the thing which acts upon an object to produce another object is called a map or morphism.

Morphisms can be composed. In the first example the tray can be lifted L and then rotated R. Composition simply means that two actions such as L and R can be thought of as combined into a single action R∘L. The symbol ∘ denotes composition.

Morphisms are associative. Think of three motions of the tray.

L: Lifting of the tray 10 cm above the table.

R: Rotation of the tray 180 degrees clockwise.

S: Shifting of the tray 1 m north while maintaining the elevated position.

The lift and rotation can be thought of as combined into a single motion followed by the shift; this is denoted S∘(R∘L). Alternatively, the rotation and shift can be thought of as a single motion following the lift: (S∘R)∘L. Associativity simply means that S∘(R∘L) = (S∘R)∘L.

An identity motion is any motion which brings the tray back to a starting position. If M denotes lowering the tray 10 cm then M∘L is an identity motion. The identity rule in the formal definition of a category states that any action preceded or following by the identity is equal to the action alone.

This formal definition embodies the preceding concepts in concise mathematical notation.

... Peter Lyall Easthope 12:40, 19 May 2008 (CDT)

Hi, this is a great improvement. It still needs some work, I expect, but you're now in the right ball-park (or cricket-grounds, depending on which side of the Atlantic you're from :-). A couple of suggestions:

• Lose the "High school mathematics" example; it doesn't add much, and basically just slows down getting to the text about what category theory actually is.
• The section about "The mathematical abstraction" could probably use a little more expansion. For example, I am getting the impression (and maybe this is incorrect, if so, apologies) that a 'category' consists of a set of objects, along with a set of maps/morphisms that can be applied to that set, and that category theory allows one to say something about anything which meets that definition? If that's sort of correct, something like that (with any errors I have included fixed, obviously) would be a useful thing to add there.
• Some text explaining what the importance of category theory is, and what it is used for (i.e. the kinds of problems it can be used on), and how it is used, would be really useful and informative.

Anyway, you're getting there! J. Noel Chiappa 13:28, 19 May 2008 (CDT)

So now which is better: edit the essay where it is or tag the fresh version on here? ...Peter Lyall Easthope 10:17, 20 May 2008 (CDT)

Oh, just go ahead and transplant the text into the base page of the article, and work on it there. That's the usual mode of doing stuff here; it's marked as a draft article that's just getting started. J. Noel Chiappa

I always thought that category theory appeared because of totally different reasons. Burbaki have shown the important conception of structure in mathematics. But there are still too many structures and a lot of similiar definitions in them. The category theory reduced similiar definitions for different structures to one definition by using maps between objects. That's how a "metastructure" of category that contains only the objects and maps between them appeared. Andrey Khalyavin 03:48, 21 May 2008 (CDT)

Andrey, What you say is correct, as I am aware. As Noel has suggested, the introductory section should aim to explain the idea to someone with a high school mathematics background. References to structures, metastructure and maps will lose at least 99% of readers at that level. The evolution and role of category th. in mathematics should be an addtional section in the article. Draft it if you are interested. The story can begin before Bourbaki; Cantor or perhaps earlier(?). There are categories in physics as well, although few documents address this. Regards, ... Peter Lyall Easthope

Noel, thanks for your ideas. I've just moved the prototypical introduction into the article with the first two edits you suggested. Your third suggestion is merely two new headings for now. Ideally the Role section should be drafted by a professional mathematician. Are you interested Andrey or Giovanni? I can tackle the Evolution section if there are no other volunteers. Peter Lyall Easthope

As of 2009-05-24, terminology is beyond "beginning college math". The introduction, for instance, refers to "intuitionistic type theory" and "functional programming semantics". "The canonical example" should be understood by beginning college students but many will have difficulty. Also, Cat. Th. is central to pure mathematics and has applications in all disciplines of science. Emphasis on computer science can be excessive. Regards, Peter Lyall Easthope 00:53, 25 May 2009 (UTC)

I don't think a single sentence can be called excessive. I just put the only non-mathematical application I'm actually familiar with.Ashley J. Ballard 21:52, 1 June 2009 (UTC)

Examples

(copied from Talk:Category theory/Related Articles)

Jitse & others, I notice that under Examples [on Category theory/Related Articles] we now have "Category of sets" and "Set". Set, in boldface, is the name for the Category of sets. On the other hand, a set alone is not a category. So the first two items would properly be stated as one example of a category. Likewise for "Category of schemes" and "Scheme". The list of examples needs tidying.
Regards, ... Peter Lyall Easthope 19:02, 1 September 2008 (CDT)

I struggled a bit with what to put there. The problem is that the link to Category of sets is not very useful because the article does not yet exist. That's why I also put in the link to set. However, that article is about individual sets and has no discussion about category-theoretical aspects, so perhaps that link is not very useful either.

I can think of the following possibilities:

1. Category of sets [r]: Category whose objects are sets and whose morphisms are functions between those sets. ^[e]
2. Set [r]: Category whose objects are sets and whose morphisms are functions between those sets. ^[e]
3. Set, the category of sets [r]: Category whose objects are sets and whose morphisms are functions between those sets. ^[e]

I'm leaning towards the second possibility; in the description we can explain that the objects in the category Set are sets. If required, all three of them can be combined with

• Set [r]: Please do not use this term in your topic list, because there is no single article for it. Please substitute a more precise term. See Set (disambiguation) for a list of available, more precise, topics. Please add a new usage if needed.

but I gather you would rather not do this, for the (very sensible) reason that a set alone is not an example of a category. What do you think is best? -- Jitse Niesen 06:05, 21 September 2008 (CDT)

The second instance is best to me also. A possible description is "The category of sets and functions.". There would also be categories with sets as objects but with maps other than functions; but I can not cite off hand. Thanks, Peter Lyall Easthope 10:01, 21 September 2008 (CDT)

Category 2

Folk,

A description of the category 2 is now visible at http://carnot.yi.org/Category2.xhtml . It passes http://validator.w3.org/ and should be visible with Firefox, Iceweasel and Vista. Many older browsers will display the document badly or not at all. The blue grid will disappear if GridColor is set to "none".

I need help as follows.
(1) Debian Iceweasel displays this OK. Will someone with access to a Vista system please open the document with the system browser. Please make a jpg screencapture and post it on a server or in the CZ image archive. You should be able to do that with File>Export or with a camera. Please notify peasthope.at.shaw.ca when the jpg is available. I'll endeavour to fix graphical defects displayed by Vista.
(2) Suggestions for improving the source text of the document from anyone with expertise in XML are welcome.
(3) Suggestions for improving the description from anyone with expertise in Cat. Theory are welcome.
(4) When there is a concensus that the document is ready, I'll need help to install it as an example in the Category Theory article.
Thanks, ... Peter Lyall Easthope 19:04, 6 November 2008 (UTC)

Seems that nobody has noticed this. I am making a link hoping that it gets some attention.
Peter Lyall Easthope 01:57, 16 November 2008 (UTC)

Location of the examples

The examples are now in the Related Articles. Will anyone object if they are deleted from the main article? Regards, Peter Lyall Easthope 15:36, 4 December 2008 (UTC)

Tray example

I have changed the example of the tray on the table so that it is no longer lifted and lowered. This is because if the tray is on the table it cannot be lowered any further. So whether the action of lowering can be done depends on where the tray is, so the "objects" are not the tray, but the possible positions the tray can be at. I think this would be confusing to readers if explained; better to just keep the tray on the table and move it around freely.Ashley J. Ballard 13:33, 23 May 2009 (UTC)

Hello Ashley, the tray example was really taken from linguistics rather than from kinematics. A noun was an object and verbs were maps. My description should have been more clear. I chose the tray example and the calculator example because they would be understood by many high school students. Incidentally, some authors use the word morphism and others prefer map. For an unsophisticated reader, map is less intimidating. Regards, ... Peter Lyall Easthope 01:09, 25 May 2009 (UTC)

I agree about "morphism". My worry with "map" would be that it is so easily confused with function. The other possibility is "arrow". How about that?Ashley J. Ballard 21:52, 1 June 2009 (UTC)

The definition of a category

For novice readers a heuristic definition before a formal definition can be a great help. As of 2009-05-24 the formal definition begins immediately after the introduction. I am not sure that the definition stated by Ashley J. Ballard is better than the definition written by Giovanni Antonio DiMatteo with minor revision by Andrey Khalyavin. This question should be judged by a few mathematicians. Regards, ... Peter Lyall Easthope 01:27, 25 May 2009 (UTC)

This definition is essentially the one at the start of MacLane (1997). I think it is better because it does not make any commitment on what the foundation is, whereas the previous one was very set-theoretic. It also does not use the vague term "class". Authors who talk about classes then have to either leave it undefined or use one of the non-standard set theories that deal with classes, or add axioms to ZF to ensure the existence of universes. None of this is necessary and I don't think it belongs at the start of an intoductory article on category theory.Ashley J. Ballard 21:52, 1 June 2009 (UTC)