Talk:Category (mathematics)

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 Field: Foundations, logic, and set theory

Page name[edit]

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)

Subcategories of Set?[edit]

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)

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

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)

Basic Category Theory for Computer Scientists[edit]

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)

Composition is a binary operation?[edit]

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 × SS, not to a function S × RT. 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 × SS. Then any function A × BC would be a partial binary operation: just take S = ABC...! Jayme 16:58, 21 April 2007 (UTC)

Ens[edit]

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?

why to use small categories?[edit]

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)

category definition of real numbers, if available[edit]

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)

How about: Reals = class of all complete, ordered fields with field-isomorphisms between them? Thus, Dedekind cuts, equivalence classes of Cauchy sequences, and other models of the Reals are merely the objects of the category: Pick any one of them if you want to work on the internal details; but what makes the Reals the Reals is what all those models have in common -- which exactly what the stated class and its morphisms describe. Just a suggestion. Jmacwiki (talk) 16:39, 3 December 2009 (UTC)

Recent revisions to the definitions[edit]

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)
Sam Staton is right (and a professional). A collection of objects and a collection of morphisms, together with the operations 'source' and 'target', is simply a directed graph. A category is, by definition, such a graph equipped with composition and identity operations satisfying associativity and unit axioms. A single graph can carry many different category structures. You have to specify which one you want. 86.156.166.193 (talk) 02:14, 8 March 2009 (UTC)

I removed the statement "The morphisms of a category are sometimes called arrows due to the influence of commutative diagrams." I don't believe there is any evidence for the "influence of commutative diagrams". When I used it in my writing (for example, Michael Barr and Charles Wells, Category Theory for Computing Science third edition. Les Publications CRM, Montreal, 1999.) my motivation was to avoid the suggestion that "map" and "morphism" give that an arrow has to be defined on elements. But others who use the word may have other motivation. Better not to say anything. Wellsoberlin (talk) 19:46, 10 April 2009 (UTC)

Cat category?[edit]

Why is the category of categories and functors a small category? Or even locally small? What aspect of the stated definition fails to cover the case of all categories and functors? Jmacwiki (talk) 16:44, 3 December 2009 (UTC)

Cat is not small and the "categorie of all categories" (in the sense described in this article) does not exist, since the class of all classes does not exist.Stephan Spahn (talk) 12:36, 11 May 2011 (UTC)
Well, either I misread, or the article has been fixed to remove any statement that Cat is small. Either way, my confusion is reduced.
As for it not existing: The set "S" of all sets does not exist, as those words are conventionally understood. That's because we can prove that it doesn't: The generally accepted axioms of set theory (ZFAC, presumably) require that the intersection of any set [hypothesized to be S, here] with any class C [of all things having the property that they do not contain themselves as members] be a set; only it isn't, because of the contradiction. So S itself must not exist.
I do not recall that the axioms, or others that are generally accepted, impose the same requirement on classes. (Indeed, that seems to be the value of distinguishing sets from classes in ZF.) Do they? Jmacwiki (talk) 01:20, 30 May 2011 (UTC)
Well, there is the notion of Cat which is the category of all strict small categories. See Cat in nLab for details. Seems we also have an article here on WP: Category of small categories. And, by appealing to the idea of a 2-category to hold them all (the enriched category over Cat), aka higher category theory, you can build the 2-category of all categories, without contradiction. Or so they say; I don't yet understand it. linas (talk) 04:23, 21 August 2012 (UTC)

Does hom() mean homomorphism?[edit]

In the Definition section, it's implied that hom(a,b) means "a morphism from a to b". So first of all, is this is a correct supposition?

One reason the article leaves this in doubt is because "hom" is conspicuously similar to "homomorphism", which in turn is conspicuously different from plain "morphism". So a reader (ie: me) would be resistant to jumping to conclusions about which one is actually the same as (or a superset of) the others. For example, would it be correct to infer that "homomorphism" is a synonym for "morphism"?

Anyhow, it would be very helpful for these basics to be spelled out. Gwideman (talk) 01:11, 15 December 2009 (UTC)

Hom(a, b) is the collection of morphisms, not an individual one.
Yes, it's short for homomorphism. I don't think there's a fundamental difference. It's just that in some contexts you talk about homomorphisms, while morphism is a more general term. Peter jackson (talk) 15:03, 25 May 2010 (UTC)

Rel and allegories[edit]

The category Rel consists of all sets, with binary relations as morphisms. Abstracting from relations instead of functions yields allegories instead of categories.

This is very confusing. I think the underlying problem is that "allegories instead of categories" implies, falsely, that allegories are not categories. 68.239.116.212 (talk) 05:31, 13 January 2010 (UTC)

"Between" two objects[edit]

Any preordered set (P, ≤) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x ≤ y. Between any two objects there can be at most one morphism.

Is "between any two objects" ambiguous? If there is one morphism from A to B and one from B to A, is the condition "between any two objects there can be at most one morphism" violated? Not according to the meaning it must have in the above, but one might interpret it otherwise. How about saying "There can be at most one morphism from any object to any other object"? 68.239.116.212 (talk) 06:05, 13 January 2010 (UTC)

References[edit]

I find it strange that so many basic facts are attributed to Jacobson. I think it is normal to acknowledge "Categories for the Working Mathematician" as the standard category theory text. I was thinking of changing the references from Jacobson to Mac Lane. But I actually don't think that all these individual references are necessary. I don't think we need 13 separate references all to the same 4 pages of one book. It would be better to start the article with "A standard text for this subject is CWM. All the concepts in this article can be found in the first few chapters of that book or of the other textbooks at the end of this article." Any strong feelings, before I change anything? — Preceding unsigned comment added by ComputScientist (talkcontribs) 15:51, 12 March 2011 (UTC)

I do think it is necessary to have those inline references, even if they are not from Jacobson (which was just the book I had at hand at the time I've added them). That said, I'm not convinced that is was an improvement removing the references without replacing them with some other more appropriated citations. Helder 17:48, 28 March 2011 (UTC)
Hi Helder. Do you think that inline references are necessary because they are helpful to the reader? Or because it is a mechanism for ensuring the accuracy of wikipedia? If yours is the latter reason, then I don't think any one is disputing the correctness of these basic statements. As for the first reason, I don't see why it is helpful if every single sentence has an inline citation to the same chapter of the same book, even the same page. I have put a note at the top, saying that everything is in the first few chapters of the textbooks, and in fact it is all within Chapter I of CWM. If there is anything that is not as well known, an inline citation would be fine. Or perhaps you can think of a different compromise. ComputScientist (talk) 08:19, 29 March 2011 (UTC)
Hi!
Both reasons, but with varying importance depending on the reader. For experienced mathematicians, I do agree that citations for basic facts are not really necessary, because they are familiar with those ideas and won't dispute any of those statements. For these readers the most important references will be those related to deep results (e.g. references indicating a text where a proof of some theorem can be found). On the other hand, for readers who are still learning the basic facts and consults Wikipedia to get a general picture of the subject, even citations for specific examples can be very helpful to provide references for further reading, so that the student can have a better understanding of the ideas (admitting that Wikipedia articles are not supposed to present its content with as much detail as is found in a text book, like those from Wikibooks - my "home wiki" ;-) - in which the didactics is of greater importance). The references can also be used by other editors as a source of information (for further improvements in the article).
Usually, I prefer to adopt an approach which is closer to Wikipedia:You do need to cite that the sky is blue than it is to Wikipedia:You don't need to cite that the sky is blue, but if it is needed we can merge the references which points to the same page of the book (there were 3 pointing to page 11 of the book and other 6 to page 12). Helder 20:34, 29 March 2011 (UTC)

Lead[edit]

Hi. In March I put the category of sets in the second sentence of the article, because I believe it is the most important and informative example of a category. I've reinstated it there. It's true that monoids and preorders are categories, and that is often useful, but that's not why categories were invented and it's rarely the primary motivation for them. Still, I've left them in the third sentence in case Classicalecon feels strongly about it. Best, ComputScientist (talk) 09:57, 3 June 2011 (UTC)

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Element-free definition needed.[edit]

Somewhere toward the end of this article, we should provide several alternative definitions for a category, including an element-free one. viz using diagrams. In particular, it would be nice to show how a category is a 3D simplicial set with objects being 0-D, morphisms 1-D, composition 2D and associativity 3D. I think that's a slick trick worth showing. Also at least a short blurb about enriched categories and other extensions... linas (talk) 04:41, 21 August 2012 (UTC)

I have no clue why I felt I needed to write the above. Its just another random idea, a to-do list item. linas (talk) 04:46, 21 August 2012 (UTC)
Hmm, the article on enriched category has several figures that could be recycled here... linas (talk) 05:28, 21 August 2012 (UTC)

Large and Locally Small Categories[edit]

I think that all the examples you have given of large categories are locally small.

An example of a large category that was not locally small and stated as such would help make the distinction clearer. — Preceding unsigned comment added by 86.27.207.97 (talk) 23:04, 22 February 2013 (UTC)