Talk:Category of sets

Nullary intersection

Here is a beginner question that should be addressed: notable in its absence is any mention of the idea of intersection of sets. I presume that this is intentional, and due to the problem of the nullary intersection. This should be belaboured.

In a related sense, the empty set cannot be a terminal object; I think this point could be belaboured. That is, the reason that the empty set cannot be terminal is because there do not exist any functions that do not map anything: a tautologically vacous truth. This should not be confused with the case of a function that maps all the elements of a set to the empty set: such a function does exist, and it is an example of a morphism to a singleton: in this case, the singleeton is the set containing the empty set. Similarly, all singletons are terminal objects.

A discussion along these lines would benefit the article, but I'm not feeling bold enough to do this myself. linas 01:24, 21 July 2006 (UTC)

Intro bit

"In mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics."

It seems ambiguous to me. Perhaps it should read "the category whose object class is the class containing every set and whose morphism class is the class containing every function". When I first read it, it seemed to be saying "every object in the category is a set", etc.

I can't edit that bit, so this is my two cents. Dissimul (talk) 09:20, 18 December 2007 (UTC)

Introduction; Examples

I agree with linas and Dissimul but am not anxious to change the first two sentences. Examples might help with some of the concepts.

What about this wording?
"... is the category comprising all sets as objects and all functions as morphisms." Is "class" necessary?

d> ... seemed to be saying "every object in the category is a set" ...

True isn't it? Is there an object in Set which is not a set?

What about mentioning some of the objects explicitly?
The empty set: {}.

Sets of one element: {{}}, {0}, {1}, {2}, {(0,0)}, {(0,1)}, {(0,0,0)}, {a}, ... .

Sets of two elements: {{}, 0}, {0, 1}, {0, 2}, {0, (0,0)}, {(0,0), (0,1)}}, {a, b}, ... .

Sets of infinitely many elements: the natural numbers, the reals, ... .

Regards, ... PeterEasthope (talk) 05:04, 26 January 2008 (UTC)

I think the examples are really unnecessary and distracting here. One should go to the set article to see examples of sets. This article should focus on categorical properties of Set rather than on sets themselves. I also don't think there was any problem with the old wording in the opening sentence. It reads better to me the way it was. -- Fropuff (talk) 19:08, 29 January 2008 (UTC)

An object is a constituent of a category rather than a possession of it but I've reverted it, ...PeterEasthope (talk) 19:25, 29 January 2008 (UTC)

I see that—thank you. I didn't mean to sound harsh here. I know your edits are well intentioned. You are correct in your assessment above that every object in Set is indeed a set. And yes, it is necessary to point out that the class of all sets is proper. So the word class is necessary at some point. -- Fropuff (talk) 19:39, 29 January 2008 (UTC)

On the "class" issue: this depends on your foundational standpoint. Some people, including Mac Lane in his book, propose to assume an axiom of universes in the ambient set theory, rather than a set/class distinction. Also, Russell's paradox doesn't necessarily prevent a set of all sets, and there are different foundations that allow different things. I propose to truncate this paragraph, just leaving the fact that it is a large category and leaving a proper discussion to (say) the Large category article.

Regarding the constituent versus the possessive: I agree with Peter that "whose" is a bit ugly, although often used. An alternative wording is "In mathematics, the category of sets, denoted Set, is the category with objects all sets and with morphisms all functions.". This is closer to Mac Lane's wording, and not far off the current wording here... How does it sound? In any case, if someone doesn't know what Set is, they probably shouldn't be looking at this page: they'd be better off looking at the more introductory articles on category theory, and perhaps should be redirected there. Sam Staton (talk) 22:42, 29 January 2008 (UTC)

On the contrary, I think this article is a perfect place to address foundational issues. I encourage you to elaborate on these ideas in the article. The category of sets is the prime example of why we want to consider large categories, and this needs to be mentioned. We should mention Mac Lane's viewpoint here as well with a mention of the category of small sets (objects are sets living in a chosen universe). This is a small category.

Regarding the possessive: It seems natural to regard a category as something possessing two classes (the class of objects and the class of morphisms). In turn the class of objects possesses those objects. It then is correct to say that the category of sets is one whose objects are sets (that is, whose object class is the class of all sets). Grammar aside, I just think it reads better. -- Fropuff (talk) 23:24, 29 January 2008 (UTC)

Sam, Fro & others,

I prefer your wording Sam.

Nobody has argued the other grammatical issue, singular versus plural. Where possible, singular should be used. It isn't just a formality. "Every" focuses attention on one at a time. "All" requires the reader to focus on the totality first. Hence, "every set" is better than "all sets".

Many students are familiar with sets and can easily give examples. Even so, the concept of "all sets" and "every set" is difficult for a new student of categories. The two words "all sets" can be read so easily without appreciating the vastness they address. My objective with the examples was to help with this difficulty. Redirection to another article might also help. Would you allow an additional sentence such as "If something is a set, it is in Set; anything not a set is not an object in Set." Are there other ways to get the idea?

You fellows are building a beautiful edifice. I admire it. Now there should be ways to make the exposition powerful as well.

Regards, ... PeterEasthope (talk) 03:36, 30 January 2008 (UTC)

I have no objection to dropping the word "all" in the opening sentence as long as this point is clarified later. However, the additional sentence you propose seems unnecessarily pedantic to me. I really don't think people have a problem considering the collection of all sets (even if they don't grasp the full significance of it). Russell's paradox is so named because it seems counterintuitive that such a collection should not be a set. -- Fropuff (talk) 04:40, 30 January 2008 (UTC)

f> ... dropping the word "all" ... clarified later.
Right oh; but, at present, I have no sound idea for the clarification.

f> ... Russell's paradox ...
That paragraph fits well.
Regards, ... PeterEasthope (talk) 19:42, 31 January 2008 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Category of sets/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Utility of the categorical view needs to be discussed more. CMummert · talk 19:07, 11 May 2007 (UTC)

Last edited at 19:07, 11 May 2007 (UTC). Substituted at 01:51, 5 May 2016 (UTC)