Timeline for Why need the morphisms to form a set ?
Current License: CC BY-SA 2.5
25 events
| when toggle format | what | by | license | comment | |
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| Dec 10, 2010 at 8:48 | comment | added | Ryan Reich | Here's one we all should have found: mathoverflow.net/questions/3278/…. The accepted answer is my example (apparently they are called "spans") and Qiaochu's and David's examples show up in the next two answers. | |
| Dec 10, 2010 at 0:07 | comment | added | José Figueroa-O'Farrill | Related question: mathoverflow.net/questions/39073/… | |
| Dec 9, 2010 at 22:48 | comment | added | Mariano Suárez-Álvarez | The category of functors between two categories, with natural transformations as morphisms, tends to have big homs without some hypothesis (a set of generators, or something like that), no? | |
| Dec 9, 2010 at 22:14 | comment | added | Kevin Buzzard | It didn't annoy me---but the first time I saw the question all there was was the question (with no explanation as to why it was being asked) and a couple of answers that just said "look up the definition of small/locally small". My reaction was that I still needed convincing that this was a question worth asking. But the convincing came very shortly after I asked :-) That's one of the neat things about this site. On the other hand, Ralph, you could have put your comments in the question---that way it would have looked less like a "what happens if I choose an axiom at random and drop it" q:-) | |
| Dec 9, 2010 at 21:55 | comment | added | Ralph | @Kevin: The motivation behind the question was: With the hom-set definition the functor "category" $C^I$ generally requires I to be small in order to be a category itself. This restriction looks somewhat artificial. Or vice versa: If it were possible to drop hom-sets than the functor category is always a category, which seems natural to me. And I'm sorry if my question annoyed you somewhat. | |
| Dec 9, 2010 at 21:44 | comment | added | BCnrd | Dear Kevin: this issue really arises when defining derived categories via localization as in David Roberts' comment just above. See 10.4.4 in Weibel's book, for instance. (In fact, Weibel should have "locally small" hypotheses in many places where it is omitted.) Also, when Grothendieck proves his criterion for an abelian category with a "generating object" to have enough injectives (via various axioms called things like AB1, AB2*, etc.), he really uses transfinite induction on Hom sets in a clever way. So there are reasons other than Yoneda. | |
| Dec 9, 2010 at 21:43 | answer | added | Chris Heunen | timeline score: 4 | |
| Dec 9, 2010 at 20:55 | comment | added | David Roberts♦ | @Kevin Perhaps another example that will seem less artificial is the localisation of a non-finitely complete, non-small category $C$ at a class $W$ of arrows not having a category of fractions a la Gabriel-Zisman. A priori this is not locally small, since the arrows of $W^{-1}C$ are (classes of) zig-zags of arbitrary length, with the wrong-way arrows in $W$. I can't quite recall if Quillen uses it for non-small categories, but this is how he defines the fundamental groupoid of a category (taking $W = Arr(C)$), and this is certainly an interesting question for me. | |
| Dec 9, 2010 at 20:54 | comment | added | Kevin Buzzard | @Ryan, David, Qiaochu: together you have convinced me that this is a genuine question. Nice one :-) | |
| Dec 9, 2010 at 20:47 | comment | added | Kevin Buzzard | @David: I am interested to hear what use Ryan's category has. I can see the merits of 2-categories because I see them in algebraic geometry when considering moduli spaces. But I don't see Ryan's example showing up in what I do. Do people studying simplicial sets or infinity-categories or something ever really see this 'category'? | |
| Dec 9, 2010 at 20:45 | comment | added | Kevin Buzzard | So it seems to me that in fact a good answer to this question is perhaps something like the following: "This can happen! It happens when the objects of your 'category' are too big to be sets, for example, if they are categories themselves. But fortunately what happens, in this situation at least, is that the 'category' inherits some extra structure (that of a 2-category), and the "set-ness" is still there---not in the hom-sets but somewhere a bit deeper, so one has to dig a bit deeper." | |
| Dec 9, 2010 at 20:44 | comment | added | David Roberts♦ | Ryan's example is not artificial in the absence of Choice - and unless you have a set of projective covers $C \to A$ for each $A$, this is most definitely not locally small. I think the question has merit, even if it needs a slight rephrasing. I read the OP to be asking something like 'I've heard the phrase "locally small category". What's the distinction to categories as usually defined (with hom-sets) and why is it important?' | |
| Dec 9, 2010 at 20:40 | answer | added | David Roberts♦ | timeline score: 10 | |
| Dec 9, 2010 at 20:39 | comment | added | Kevin Buzzard | @Qiaochu: the most natural objects that I can think of that show up in mathematics and that are not sets, are categories. So maps between categories do form a natural example. | |
| Dec 9, 2010 at 20:38 | comment | added | Kevin Buzzard | @Qiaochu: that's not a category! That's a 2-category. But I'm definitely much more excited about that example than about Ryans. | |
| Dec 9, 2010 at 20:37 | comment | added | Kevin Buzzard | [PS I thought that a multifunction $A\to B$ was just a function from $A$ to the non-empty subsets of $B$; such things of course form a set, but you seem to be using the word in a much more general context] | |
| Dec 9, 2010 at 20:37 | comment | added | Qiaochu Yuan | @Kevin: what about the category of categories? That's certainly a non-artificial example. | |
| Dec 9, 2010 at 20:33 | comment | added | Kevin Buzzard | So, in particular, I am asking that "I can't say confidently that it is useless" hopefully be replaced by "I can say confidently that it is not useless", which is asking much more, and, I think, the minimum that needs to be asked to make this question of interest to research mathematicians. Note also that I am definitely not saying "no such example exists"---but I am saying "one really needs an example that actually happens before one can justify thinking about this question". | |
| Dec 9, 2010 at 20:33 | comment | added | Kevin Buzzard | @Ryan: I don't buy it. I'm not just asking "construct a 'category' with proper-class hom-sets"---that's easy, just as it is easy to construct e.g. a 'group' which doesn't have inverses. I'm asking "give me an example where such a thing really happens in mathematics". If no such example can be found, I claim that this question should be closed as not being of interest to research mathematicians and being nothing other than something that makes syntactic sense but not much more. | |
| Dec 9, 2010 at 20:11 | comment | added | Ryan Reich | Here's a "category" with proper-class hom-sets. The objects are all sets; for any two sets A and B, Hom(A,B) is the class of all sets C together with a surjection to A and a map to B. In other words, multifunctions from A to B. I don't have a use for it but I can't say confidently that it is useless and it is certainly not that artificial. | |
| Dec 9, 2010 at 20:07 | comment | added | Kevin Buzzard | What I am saying is this. Take any mathematical object, defined by a list of axioms. Choose of the axioms and then ask on MO "what is the point of this axiom anyway? Why can't I drop it?". I am not so sure that this is a very good way of generating MO questions! But of course some of these questions are good, the answer being "if you drop this axiom then you recover the notion of a (blah), and these are widely used in (blah)". Is this question really one of the good ones? If so then there will probably be some natural situations where the axiom fails. Where are these situations?? | |
| Dec 9, 2010 at 20:03 | comment | added | Kevin Buzzard | Dare I ask what the motivation behind this question is? I can't imagine that a research mathematician would actually run into this issue in their research. If the objects of your category are any sort of reasonable thing (e.g. if they remotely resemble sets with additional structure) then the hom sets will automatically be sets. | |
| Dec 9, 2010 at 19:47 | answer | added | Ryan Reich | timeline score: 8 | |
| Dec 9, 2010 at 19:42 | answer | added | Adam Hughes | timeline score: 6 | |
| Dec 9, 2010 at 19:31 | history | asked | Ralph | CC BY-SA 2.5 |