Talk:Category (mathematics)

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: B Class Top Priority  Field: Foundations, logic, and set theory

Contents

[edit] Page name

I moved this page from category (category theory) back to category (mathematics) as this is the most common usage. The other usage is category (topology) which is usually referred to by Baire category, first category, or second category. -- Fropuff 17:31, 2005 Jun 1 (UTC)

[edit] Subcategories of Set?

The examples given, Rel through Uni, don't appear to be subcategories of Set since generally a set can be made into, for example, a group in many nonisomorphic ways. I think the original author meant to point out that they are concrete categories over Set. Moreover, Rel doesn't even appear to be concrete, since many relations are not functions. Is there some way of viewing these categories as subcategories of Set that I'm not seeing? If so, it should probably be noted, even if not explained in detail. SirPeebles 03:25, 26 December 2006 (UTC)

I'm writing to concur with SirPeebles; Rel is a supercategory of Set, not a subcategory. Also, the start of the section promises to describe composition in each example, but this is not done for Rel. 66.32.207.31 20:33, 12 March 2007 (UTC)

I removed the following offending sentence:
(The following are subcategories of Set, obtained by adding some type of structure onto a set, by requiring that morphisms are functions that respect this added structure, and where morphism composition is simply ordinary function composition.)
I agree they don't seem to be subcategories; perhaps the original author wanted to say "there is a forgetful functor from these Set", which is not the same thing as saying they're subcats. However, it seems premature to try to talk about forgetful functors so early in this article, and so it seems better not to say anything at all. Err, I take that back; this article is notable in failing to use the words "concrete category", and so perhaps a few sentences to that effect should be written, and the list properly classified. linas 04:16, 14 April 2007 (UTC)
Done. Cruft patrol. Some editor unthinkingly slapped Rel at the top of the list. I simply changed "subcategory" to "concrete category" and so all should be well. linas 04:49, 14 April 2007 (UTC)

[edit] The term "hom-class" in the Definition

The Definition contains this phrase. "... denote the hom-class of all morphisms from a to b." but hom-class is not a link to a definition. What about defining hom-class in the article on Class (set theory) and linking to that? Regards, ... PeterEasthope 18:58, 19 February 2007 (UTC)

That sentence is attempting to define the phrase "hom-class" as the "class of morphisms"; that's all that it is. "Class of morphisms"is a mouthful, so its just "hom-class" for short. linas 03:56, 14 April 2007 (UTC)

[edit] Basic Category Theory for Computer Scientists

I was reading Basic Category Theory for Computer Scientists (Pierce, 1991) today, and the intro to the first chapter seemed... to remind me of something. After doing some history searching, it seems that anon user 63.162.153.xxx wrote the Category theory article from scratch, and that text was eventually broken up to create this article. Problem is, that the definition of a category that was used is very, very close to directly lifted from Pierce's book.

I don't want to suggest that the whole article is a copyvio, but it certainly would be good to re-write the definition section in new language that doesn't duplicate this text. -Harmil 19:10, 21 February 2007 (UTC)

I take that assertion of blame back, and appologize to whoever that anon user is. The history link only shows edits back to that revision, but the article's origin is actually not available in the history. If you keep clicking on the "older version" link, you eventually get to the automated conversion. So, we may never know who put the text in there, but we can certainly re-write it. -Harmil 19:18, 21 February 2007 (UTC)
Sigh. Can you be more specific? I don't have a copy of Pierce's book. But I can pick up, for example, Rotman "An Introduction to Algebraic Topology" and on page 6, there is a definition of a (locally small) Category that is very similar to that in this article. It differs only in punctuation, and misc "filler" words that don't change the flow. I doubt Pierce copied from Rotman's older (1988) book, but from this distance, the definition of a category seems very generic. Is it really verbatim, or just logically similar? linas 03:43, 14 April 2007 (UTC)

[edit] Composition is a binary operation?

The article says "binary operation hom(a, b) × hom(b, c) → hom(a, c) called composition of morphisms". I think that we only call binary operation to a function S × S → S, not to a function S × R → T. Maybe we could call it function (but hom(a, b) × hom(b, c) and hom(a, c) need not to be sets; I don't know if an association between an unique element of B with each element of A is also called an function from A to B when A and B are proper classes.) Jayme 12:33, 12 April 2007 (UTC)

The article binary operation seems to define something called an "external binary operation" that is closer to what you want. But all this seems to be quibbling anyway. I don't believe there is any grave error committed by saying "composition is a binary operation", irrespective of whether the things being composed are classes. Right? linas 03:51, 14 April 2007 (UTC)
I agree with you. About binary operations, I believe that there is a notion of "partial binary operation" that would be a function S' ⊆ S × S → S. Then any function A × B → C would be a partial binary operation: just take S = A ∪ B ∪ C...! Jayme 16:58, 21 April 2007 (UTC)

[edit] Ens

In MacLane's book, there is repeated mention to a category Ens which seems to me like elements of a power set along with endomorphisms of the original set as arrows among the correct elements of the power set. Does this seem right?

[edit] why to use small categories?

Cone_of_a_functor says:

Let J be a small category and let C^J be the category of diagrams of type J in C (this nothing more than a functor category).

Is there a typical thing I'd like to do with a category, but can't if it is large. Or specific. How about the above article. Can't I define the category of type J in C if J is a large category?

Thanks, JanCK 11:43, 21 October 2007 (UTC)

In between I read Category_of_small_categories
the category of small categories, denoted by Cat, is the category whose objects are all small categories 
The category Cat is itself a large category, and therefore not an object of itself.
So what I'm supposed to get is: obj(C) of a category C is a class. So the elements have to be sets? The class article reads
class is a collection of sets (or sometimes other mathematical objects)
What kind of objects are these other mathematical objects? JanCK 12:04, 21 October 2007 (UTC)

[edit] category definition of real numbers, if available

given their importance it would be great if a definition of the reals could be given in this article. Rich Peterson4.246.233.26 (talk) 10:36, 26 December 2007 (UTC)

[edit] Recent revisions to the definitions

Hello, I have reverted the revamp to the definition made by COGDEN. The revamp to the intro was quite nice, but I'm not sure why the definitions were revised. I found the revised wording misleading. The revision implied that a category was just a collection of objects and a collection of morphisms; this is not true. By definition, a category must also have a composition operation that is associative. I think this needs to be stressed clearly in the definition: the composition and associativity are not derived properties of categories, they are part of the data. My other concern was the notation hom(a,b), where a and b are classes of objects. I've never seen that in print before, and I don't it's a fundamental notion. Feel free to discuss or rework if you like, though. Sam Staton (talk) 14:59, 21 February 2008 (UTC)

An arrow/morphism is usually defined as being composable and associative. Most of the literature definitions I've seen make it really simple: a category is a collection of objects and arrows (arrows being composable and associative). This is also consistent with the definition found in morphism. COGDEN 07:41, 29 February 2008 (UTC)
Composition is an operation defined at the level of a category. Morphisms are composable because the belong to a category. If you read carefully the definition in morphism you see that it refers to the category the morphisms belong to. In order to avoid a circular definition, composition should really be defined here. -- Fropuff (talk) 07:56, 29 February 2008 (UTC)