Talk:Category (mathematics)

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)

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)

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)

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)

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)

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?
 * Ens is short for "Ensemble" which is how to say "Set" in french. From the definitions of Ens and Ens_v on p.11 and p.12 of CWM, I take Ens to be a generic label for any category of sets, parameterized by and defined in terms of various (any) supersets V. The category Set itself is one of these, the category of "all small sets". I think the motivation for what may seem to be a roundabout or technical relation between Ens and Set is to take a lot of care with avoiding problems like defining the "set of all somethings" in these definitions, but it's probably more nuanced than that. Netrapt (talk) 00:34, 16 August 2021 (UTC)

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)

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)


 * 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
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. CO GDEN  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?
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  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?
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)

Since in category theory we speak of the morphisms of a category rather than the homomorphisms of a category, why is "hom" used rather than a term that suggests morphisms rather than homomorphisms. One possibility would be "morph". John Link (talk) 14:48, 22 February 2020 (UTC)

Rel and allegories
''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
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)

Lead
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)

Element-free definition needed.
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
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)

"Mathematical objekt"
I noticed that "mathematical objekt" redirects to Category (mathematics) instead of mathematical object. What is the reason for this? Jarble (talk) 01:47, 3 April 2018 (UTC)


 * I don't know the answer to your question, but you could change the redirect to Mathematical_object. John Link (talk) 21:22, 23 February 2020 (UTC)

"Like group" vs. "like monoid"?
Why do we start with the statement that categories are "like groups"? If we want the reader to start from something familiar, why mention the less similar "group" instead of the closer "monoid"? — Preceding unsigned comment added by Jmacwiki (talk • contribs) 22:33, 3 May 2018 (UTC)

This entire article is strange
The fact that Category here is compared to Groups and other Algebraic structures seems completely ridiculous, and it gives the wrong intuition I think for what a Category is. The correct analogy would be from Small Categories to other other Algebraic structures, since then the statement “A Small Category is a set equipped with ...” would logically map onto the other Algebraic structures in the chart at the top. As it stands it is extremely confusing why Catgory is so much more profound than group theory, because it appears that Categories are just groups but with less structure. INLegred (talk) 22:29, 6 November 2018 (UTC)

Below is a link to a video of a lecture by Eugenia Cheng that explains categories with simple mathematical examples and also non-mathematical examples. Perhaps this line of thinking could be used to make this article less strange and more accessible to a wider audience.

Category Theory in Life: https://www.youtube.com/watch?v=ho7oagHeqNc

John Link (talk) 23:22, 22 February 2020 (UTC)

Why the term "category"?
I know that mathematics uses all sorts of terms whose mathematical meanings does not correspond to their everyday meanings. But I'm still curious about the reason for using the term "category", which in everyday use seems to have a meaning close to that of "set" or "class". John Link (talk) 14:58, 22 February 2020 (UTC)

Why is this article separate from Category theory?
It seems to me that either they should be merged, or that at the very least, there should be hatnotes for disambiguation and context. It is too far out of my competence for me to do it, but something either is wrong, or incomplete. JonRichfield (talk) 04:29, 3 September 2020 (UTC)

Should the property of transitive arrows be in the group like structures table
categories need transitive arrows, but this is not written in the group like structures table. Is this because every group like structure has one? is it just a natural property of a closed property inherent in a group? I think it should be added for clarity. Glubs9 (User Talk:Glubs9:talk) 22:19 12 August 2021 (UTC)

Definition makes it sound almost as if hom(a, b) is non-empty for all a, b
> for every three objects a, b and c, a binary operation hom(a, b) × hom(b, c) → hom(a, c) called composition of morphisms; the composition of f : a → b and g : b → c is written as g ∘ f or gf.

I like the way nlab says the same thing but with morphisms instead: https://ncatlab.org/nlab/show/category#OneCollectionOfMorphisms

> for every pair of morphisms f and g, where t(f) = s(g), a morphism g∘f, called their composite

Actually, their second definition of a category is even easier for me to read: https://ncatlab.org/nlab/show/category#AFamilyOfCollectionsOfMorphisms Mateen Ulhaq (talk) 10:12, 22 December 2021 (UTC)

"Locally small category" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Locally small category and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 September 23 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 08:25, 23 September 2022 (UTC)

Not good information for me
Please note 2001:56B:3FE5:3FAD:D5A0:4D47:10FE:3E5 (talk) 01:32, 28 November 2022 (UTC)