This post is a bit of a rant.
One subgoal of the stacks project is to work through the beginnings of etale cohomology and algebraic stacks without making use of universes. Most of this is completely straightforward (and already done), and there is only one point at which you have to make an argument.
First I would like to point out that there is a completely well established (axiomatic) theory of sets, and that is ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Thus, virtually any mathematician who uses a set means the type of object that is described by the axioms of ZFC. Paradoxically it is the set theorists themselves who enjoy thinking about other kinds of sets. They like to add and substract axioms from ZFC and see what happens. It is probably for that reason that you cannot find a book that simply takes the axioms of ZFC and develops the theory of sets (if you do know such a book or lecture notes, please email me or leave a comment). So whenever you take up a book on set theory to learn something about the sets you work with every day, you have to be very careful to see whether the author has added some bizarre additional hypotheses to the theory, or works with a different axiom system. To me it seems a bit of a crime that some of the undergraduate level books do not use ZFC.
Of course, some of the most interesting results in set theory (that I know) are those having to do with consistency, etc. As an example I want to mention the result that if ZFC is consistent, then you cannot prove the existence of a strongly inaccessible cardinal in ZFC.
What is a universe? Roughly, a universe is a set X such that all the axioms of ZFC hold for the elements of X. It turns out that the existence of a universe is equivalent to the existence of a strongly inaccessible cardinal, hence cannot be proved inside ZFC (and neither can the nonexistence, actually). Thus I argue that most mathematicians use a set theory which does not have universes.
Grothendieck added the existence of universes U as an axiom so he could say “let (Sch) = the category of schemes which are elements of U”. This is somewhat convenient. For example it means that if I is an element of U and X_i is an element of U which is a scheme for all i, then \coprod X_i (suitably constructed) is an element of U. On the other hand (Sch) is not an element of U and considering it takes you outside of U. Moreover, since U is just some set, (Sch) is just some set of schemes, and there are schemes not in U. Furthermore, the topos of sheaves on (Sch) is of course a proper class and not a set at all. Finally, changing the universe gives you a different topos.
The axioms of ZFC provide many techniques for constructing large sets. How close can we come to constructing a universe U inside of ZFC? I don’t know; I’ve looked around, but I haven’t found somebody addressing this directly (likely because for a set theorist this is utterly trivial). But here is what is true; I’ll formulate this in terms of (Sch) = the category of schemes which are elements of U, because if you are reading this then you probably do not care about “large sets”. Given any set of schemes (Sch_0) you can construct a U such that
- (Sch_0) is contained in (sch),
- (Sch) has fibre products, and fibre products agree with fibre products in the category of all schemes,
- more generally you can construct U such that limits and colimits over at most countable diagrams in (Sch) exist whenever they exist in the category of all schemes and are the same,
- you can make (Sch) be closed under immersions, morphisms of finite type, morphisms which are locally of finite type such that the inverse image of an affine can be covered by countably many affines, and
- if X is in (Sch) and Y is a scheme whose “size” is at most the 2^(size of X), then Y is isomorphic to an element of (Sch).
The key is to construct U such that (3) and (5) hold; the size of a scheme X is defined in terms of the cardinality of the set of sections of O_X over affine opens and the cardinality of the set of affine opens. But now you cannot _also_ require that (Sch) is closed under disjoint unions of objects of (Sch) indexed by elements I of U, and, if you think about it for a bit, you will see that this is the only difference from the case of a universe. So, although it is clear that properties 1 — 4 imply that in many cases the disjoint union does exist in (Sch), it is just not always true!
This is the approach we chose in the stacks project. Another one might have been to construct U’s such that the disjoint unions always exist, but then you need to weaken condition (5) by quite a bit; I’m not exactly sure how much.
I still don’t understand why the Stacks project goes to all of this trouble. I’ve never heard of anyone (save perhaps Brian Conrad) who has expressed any doubt at all over the axiom of universes.
Axioms of set theory have nothing to do with “truth”. If you are not a set theorist you pick a collection of axioms and interference rules and work with those. I pick ZFC. What do you pick?
ZFC+U as in SGA/Les Dérivateurs, simply because it’s convenient for category theory, algebraic geometry, and homotopy theory. I guess my question is: if you’re going to go as far as accepting things like choice, which are much more controversial, why not accept a whole bunch of large cardinal axioms that make things a lot less painful?
I don’t have any objections per se about Tarski-Grothendieck set theory-which is basically what we’re talking about here. My problem is why do we need a “trick” to get around proper classes when we can just add proper classes-carefully defining the axioms accordingly to get around the paradoxes,of course-and this way,categorical and set theoretical constructions are both on an equally sound footing? I don’t take this to be simply a matter of practicality. Ontology is important for conversations about things in the world-and one of the main things that makes mathematics relevant is that it can be used to make precise dialogues about things in the world i.e. natural science.It’s not just a game with connectives and symbols.
At least that’s my take on it.
Fine. I’m perfectly happy with you using ZFC+U. That means that you can use whatever I prove and I cannot use anything that you prove. Moreover, if another person wants to work in ZFC-U they can use my results as well.
As I tried to indicate in the post, I do not think that it is “less painful” to work with universes. What I claim is that the amount of set theory that you have to add to make it work is trivial compared to the amount of algebraic geometry that we need to do. It is so trivial that whether you working with universes or not does not really make any difference (in the amount of work it takes to develop the theory).
Also, I think the word “controversial” to describe an axiom is a bit strange. Mathematically what is interesting is what you can use it for. Again, I believe only set theorists discuss the relative strengths of axiom systems (and they obtain some pretty interesting results doing so), but most mathematicians work with sets as described in ZFC.
I guess what I don’t understand is how the same argument can’t be used to justify the following statement: “Well if we don’t use choice at all, then all of our conclusions will hold in ZF-C”. If you’re using the fact that most mathematicians work in ZFC, then my argument about axioms being controversial can be restated as: “I’ve encountered very few mathematicians who work with ZFC sets who aren’t willing to additionally work with ZFC+U-sets”.
Yes, if I could work in ZF and obtain the same results I would, but I cannot. If you can that would be mildly interesting and I encourage you to do so. But as you say the real reason we should write foundations for the theory of algebraic stacks using ZFC is because most of mathematics is written and proved using the axioms of ZFC. Most of these papers are written by people who have perhaps never had a serious course about set theory (and don’t want to) and heard about Zorn’s lemma in an analysis course. So although they work with ZFC they never really think about it. If you told them (in some detail) that adding U means that many additional theorems are going to hold, then they might not be so inclined to allow you to add it as an axiom.
For these reasons I think your belief “I’ve encountered very few mathematicians who work with ZFC sets who aren’t willing to additionally work with ZFC+U-sets” can never be a serious argument for anything.
Are there any conditions (apart from perhaps some hypothetical proof of the inconsistency of ZFC) that would be sufficient motivation for you to strengthen or weaken your choice of axiom system (if not for the stacks project, at least in your own work)?
Sorry for the double post. I would edit my comment if I could. | I guess that like Grothendieck and Bourbaki, I see the addition of the axiom of universes as rectifying a flaw in set theory. Worrying about things like size issues seems rather artificial and completely irrelevant to the topic at hand (whatever it may be).
Is it really important that the category of presheaves on a small category does not admit a yoneda embedding, since the functor category [[A^op,Sets]^op,Sets] is not classically a mathematical object?
We can write down what it is. We can even pretend that it’s true when we’re thinking about proving things, but it’s not true for the simple reason that we can’t prove that the hom sets are honest sets. However, we’ve hit a wall simply because our definition of a set isn’t descriptive enough to determine whether or not this object (which seems as concrete to me as any other category) is a set.
It’s not a failure of our intuition, and it’s not a failure of our expectations to be confirmed. It’s a failure of our axioms themselves.
Not having universes is not a flaw of set theory at all. Mathematical theories do not have flaws. Axioms cannot have failures. It doesn’t make sense to say something like that. Why do you keep attributing human attributes to things in set theory?
If you use universes you still cannot talk about [[A^op,Sets]^op,Sets] as a mathematical object. You have to pick a universe for the first occurrence of Sets and (if you like) another strictly bigger one for the second. You still have to be careful about set theory and you still cannot ignore set theoretical issues.
Mathematical intuition is something that guides us when we try to prove something, informed by previously proven theorems and types of arguments that we have heard others talk about. Once we have an idea of how to prove something, then we sit down and work out the details. Most algebraic geometers do not worry at all about set theoretic issues while writing out details. And, actually, having done part of the exercise of setting up the theory from the beginning, I now say that they were right not to worry.
So in some sense this discussion is irrelevant, and it is only a small point of the stacks project to get also the details on set theory right. Feel free to clone the stacks project to start a parallel project which uses ZFC+U for its foundations.
I was just replying to your rant =). I’m not going to fork the stacks project over something this trivial.
I had difficult with universes this semester when I taught intro graduate algebra. Our discussion of Yoneda’s lemma was probably the least clear and least rigorous part of the semester. That was too bad because Yoneda’s lemma is a simple and useful concept, and now the students think it is some kind of voodoo. I do not know if my department requires majors to take any set theory course, much less one which discusses issues such as universes and inaccessible cardinals.
In Lang’s book Algebra he uses universal objects which is Yoneda’s lemma in disguise but much easier to parse for undergraduates. So for example you characterize the tensor product of two vector spaces (or modules) as an initial object in a certain category. Another example is the localization of a ring at a multiplicative set, quotient of a module by a submodule, etc. I’m not sure that in an undergraduate course you should go much beyond this anyway…?
Oops, I see I misread your question and you said graduate course. When I explain Yoneda’s lemma then I work with presheaves on a small category and there is no problem in formulating it (without universes). You just have to be a little careful as to when you can use it. Often formulating it in terms of universal properties is a good way to go, for example if you want to explain what it means to take the fiber product of schemes? I really think you do not need universes to explain this, or do you have an example of something you wanted to explain in a graduate course which is more involved?
I explained to the students that the Yoneda functor represents the functor from the functor category to the category of sets which evaluates a functor at a particular object. But of course the class of natural transformations between two functors is not necessarily a set, so the functor category is not a category. I explained to the students that this is not a serious problem. We just prove the precise statement (every natural transformation from h_X to F comes from a unique element in F(X)), and leave the imprecise statement as “interpretation”. But the students seemed unhappy with this. So I talked a little about inaccessible cardinals and universes as a way of making this precise.
Of course it didn’t help that the next topic was derived functors. First we had to make more than a “set’s worth” of arbitrary choices to define the derived functors. And then the “functor category” issue recurred when we defined “universal delta functor”: What does it mean for a functor to be “universal” if the functor category is not a category?
Just to clarify: of course I did circumvent these issues by working with small categories, just as you say. But then I had to justify working with a small subcategory of a category like “R-modules”. (I also used the Freyd-Mitchell embedding theorem to reduce the Snake Lemma to chasing elements.) Of course I
pointed out that our usual constructions — tensor products, Hom, direct sums and direct products — involve only a “set’s worth” of the category. And then I mentioned inaccessible cardinals and universes as a way of trying to make precise the idea that restricting to a small subcategory is compatible with these “usual constructions”.
Usually, if you want to show that something is compatible with restriction to a smaller subcategory, all you have to show is that the representing object is in it. For example if C’ is a full subcategory of C and if X —> S, Y —> S are morphisms of C’, and if X \times_S Y exists in C and is isomorphic to an object of C’, then the fiber product X \times_S Y exists in C’ and its image in C is a fiber product in C. Most cases can be handled by simple direct arguments like this, I think.
In the stacks project we sometimes try to avoid making a general argument when a case by case analysis works as well. For example, in SGA4 they mention something called an “espece the structure algebrique d’efinie par limites projectives finie” as a notion of a type of algebraic structure for which you can define pullback and pushforward along morphisms of topoi. In the stacks project we instead handle each possible type of algebraic structure that comes up by hand (but currently the proof of that lemma needs some cleaning up). And it seems to me this is a valid thing to do also for proving compatibilities between constructions in C’ and C (as above). After all, you want to discuss each of the most important constructions in a little more detail anyway.
Finally, when you formulate things in terms of universal properties you do not have to restrict to a small subcategory, so you don’t have to worry about that. (Here you do have to be a bit careful not to start throwing around functors of functors, etc!)
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Yoneda’s Lemma is a unifying psychological device; it doesn’t matter if it is a real result or not, because what matters is the principle in its proof which in turn can be applied in many settings. I care about universal structures because of how I can use them, not for the sake of an abstract super-general theory of universality. (In a similar manner, I sort of don’t even care if category theory is really a mathematical theory or just a language, since its arguments can be plugged into “real” situations, which is what matters. But I digress.) So the issue of “functor category” not being a category (since “Hom-sets” aren’t sets) fizzles away upon noting that at that level category theory (as far as I ever need it) has no content beyond linguistics and helping us to see certain constructions in a setting stripped of specificity which we can then plug into a real situation in which there’s no problem at all.
(I almost never discuss these things with anyone, since it tends to elicit bad reactions or misunderstandings. And adding a false statement to one’s theory also makes it easy to prove lots of things, so the fact that adding some axiom yields more theorems is not a convincing argument to me at all.)
This is also why I object to universes: they just promote laziness about paying attention to what is actually going on in one’s constructions (certainly that is the practical effect I see among most people who accept universes!). I think being attentive to what happens under the hood has value in terms of getting a grip on what makes one’s theory work, and I have never in my life seen any example* of a mathematical theory with substance (at least that I care about) for which universes are necessary. For instance, fpqc cohomology is an irrelevant concept, but the property that some functor satisfies the fpqc sheaf condition or that some map of schemes is an fpqc-torsor for some fpqc group scheme is entirely meaningful and useful without needing to ever get into the universe business. Likewise, for the etale and fppf theories it is obvious that universes are irrelevant, due to the finiteness aspects built into the definitions and openness of such morphisms. (To me, “big etale site” is a linguistic device; it is irrelevant whether it fits into some “general theory of sites” since cohomology relative to it has no purpose for anything I’ve ever seen, and saying that some functor is a sheaf for it has a completely obvious meaning which I can use readily in proofs without any need for universes.)
So I don’t find avoidance of universes to be a hindrance, or even a source of difficulties or complications. To the contrary, it forces me to pay attention to what underlies constructions, which in turn is very useful at some points in both understanding proofs and generating new ones that I need. I don’t think Harry has enough experience yet in the harder parts of algebraic geometry to have any basis for an opinion on this matter as it affects algebraic geometry. (I concede that maybe in the higher topos stuff or fancier parts of algebraic topology it could be relevant, but I plan to go through my mathematical life without ever needing that stuff, or at least any parts of it which “need” universes.)
Now I put an asterisk above. It refers to the fact that there is precisely one thing I care about which I’ve never seen how to handle without the universe stuff, but for which I feel there must be an elementary set-theoretic trick to deal with it (but which I don’t know, because I learned set theory “on the street”, as I couldn’t fathom sitting through a semester-long course devoted to such things when I could instead take a course in representation theory, complex manifolds, number theory, etc.). That is the issue of “bounding” the nil-thickenings in the definition of covers for the crystalline topos (as pertains to sheafifications), for which the crucial point is to find some systematic bounding mechanism which does not upset the ring-theoretic fiber product construction in a certain specific proof very early in the theory. (Johan will know the result I have in mind, so I won’t get into it here.) Johan, any suggestions on that?