A month or so ago, I got a note from Ross Street asking about my 1986 JPAA paper on representations of categories. The main point of that paper was to give a new, relatively simple proof of the full embedding theorem for regular categories. Unfortunately, I tried to get too general and got tangled in the variance, so that the argument could not be followed. Follows is my final note to Ross. The argument is really quite simple and can be described as follows: 1. Show that when @C (read script C) is regular, so is Lex(@C,Set)\op. (Lex=FL is the finite limit preserving functors. This result is really the only new thing in the paper.) 2. Adapt Grothendieck's transfinite induction proof of the existence of injectives in an AB5 abelian category to show that Lex(@C,Set)\op has enough @C-projectives = regular functors. 3. Adapt Mitchell's proof of the abelian category full embedding theorem to show that by taking a sufficiently large full subcategory @P in Lex(@C,Set) consisting of regular functors, then the evaluation functor @C --> Func(@P\op,Set) is full and faithful. That's all there is to it. I plan to post on triples in the next few days a revised version of the paper (it was, fortunately, one of the very first that was typed on a computer, so this is not too onerous). I will call it embedding.rev. Ross further raised the question as to whether a transfinite induction was too sophisticated for the classroom. I don't know exactly what level his course was, but I will say that if we never do a hard theorem, then the students will come away with the idea that there is no depth to category theory, an impression his colleagues will be only too happy to affirm. This argument of Grothendieck's is, after all, the first example that there could be deep results in categories. Michael ===================================================================== Dear Ross: I did not really try to understand my paper. I suspect it is a matter of trying to be too abstract and getting tangled in my own feet. At any rate, here is how I would do it for a class. First, since you didn't ask, I assume that you are happy with the fact that given a small category @C (think of that as script C; BTW, I never noticed how dreadful JPAA's scripts were), the category @L = Lex(@C,Set)\op has the property that given any functor F, there is a regular epi P -->> F such that P is projective with respect to regular epis in @C. Now consider a small full subcategory @P built out of choosing @C projective covers for each representable and a @C-projective cover for the kernel pair of each of the @C-projective covers of representables. Thus, for each object C, there is a parallel pair Q ===> P in @P, whose coequalizer in @L is C. Notice that Hom_{@L}(P,C) = PC, using Yoneda and taking the variance into account. There is an obvious functor F: @C --> Fun(@P\op,Set) that takes C to the functor P |--> Hom(P,C) = PC. This functor is clearly faithful, preserves finite limits and regular epis. The only question is the fullness. So suppose a: FC --> FB is a natural transformation. What naturality means is that for any d: Q --> P in @P, the square Hom(d,C) Hom(P,C) ----------> Hom(Q,C) | | | | aP | | aQ | | | | v v Hom(P,B) ----------> Hom(Q,B) Hom(d,B) commutes. Applied to an f: P --> C, this means that aQ(d.f) = d.aP(f). Apply this in the case that f is a coequalizer in @L to a parallel pair d,e : Q ===> P in @P. It says that aP(f).d = aQ(f.d) = aQ(f.e) = aQ(f).e and that means that there is a unique b: C --> B such that aP(f) = b.f. You finish the argument by observing that for any object R of @P and any c: R --> C, there is a lifting to an arrow g: R --> P and then aR(c) = aR(f.g) = aP(f).g = b.f.g = b.c.
The proof Michael Barr had in mind at the time of writing his 1986 JPAA paper on representations of categories, and has now clarified, is truly beautiful. I may have misrepresented what the problem was with presenting the transfinite induction part of the proof in my course. It is a question of time rather than difficulty. I am not known amongst my colleagues as one who shrinks from teaching or avoiding hard proofs. The point is that I have a course consisting of 24 lectures to a group including a (bright) 4th year student (who had hardly heard of categories before the course), a graduate students in number theory, physics and computer science, my own 5 PhD students, and a professional category theorist. I am trying to keep it interesting for all. The students (even the ones who have done a formal set-theory course) have not really used ordinals; so I would need to sacrifice some category theory to provide that background. I am still undecided about this; perhaps, the transfinite induction and the embedding theorem can be an appendix to the course. Perhaps the most novel aspect of the course, considering my 2-cell background, is that I have given 18 lectures so far without introducing natural transformations. I'm about to talk about adjunctions without them too. I must admit that the course content has evolved on the run. But looking back, I think it has some unity of purpose. I headed straight for the definition of regular category as having a terminal object, pullbacks, strong epi - monic factorizations, and stability of strong epis under pulling back along arbitrary arrows. We developed the method of generalised elements and constructed the Poset-enriched category of relations in a regular category. We proved relations with right adjoints are graphs of arrows. We proved strong epis are regular. Equivalence relations and (Barr-)exact categories were examined. [One of my (intended?) questions to Michael Barr was whether he knew of a diagram lemma which could be proved significantly quicker using the embedding theorem than by the generalised-element/relations technology.] I proved in detail that an exact additive category is abelian (as defined in Freyd's book "Abelian Categories" - beginning of Chapter 2). Not only is that a deep theorem of category theory (in my opinion - despite not needing a transfinite argument) but it is a microcosm of categorical ideas that have proved useful in other contexts. We proved the Five Lemma, Snake Lemma using relations in the abelian category. So far I have set 15 exercises: some quite challenging. One was to prove Cat is not regular (I gave a bit of a hint). Now I am discussing 2-sided discrete fibrations (just for categories) in depth. These are being advertised as the "relations" of category theory. In fact, this means we do have presheaf categories disguised as DFib(A,1); so some natural transformations are there hiding there. I have this weekend to decide where we go next! But I think there is enough meat in all this to deal with sceptical colleagues. When it comes to a game of "my subject's harder than yours, nya, nya", I would also argue that the concepts category theory has to offer are just as hard to really master as difficult theorems. This is why many mathematicians calculate hard to do what we can do with our concepts. Regards, Ross
I certainly welcome Ross' clarification of his problem. One possibility would be to give the proof except for the existence of sufficient regular functors and either refer the students to the paper or (better) write it out carefully and distribute it. One thing I have given only a little thought to is whether it can be done using a maximal principle argument. The point is that it is not a question of extending a map to a larger and larger subobject, but of building larger and larger objects and not as subobjects of something already given. This makes it different from the proof, say, that divisible abelian groups are injective (from which the existence of sufficient injectives in any module category follows easily). I think a maximal principle argument goes down a lot more easily than one based overtly on ordinals or transfinite induction. Ross, it sounds like a beautiful course and if you have notes, I would like to see them. Michael
Ross Street asks Michael Barr if he knows "of a diagram lemma which could be proved significantly quicker using the embedding theorem than by the generalized-element/relations technology". Well, I don't exactly, but one of the earliest uses of Barr's embeddings of regular categories was in my American Journal of Mathematics paper on Simplicial Homotopy in an Exact Category (1977, vol 99, pp 1193-1204). I wanted to prove the homotopy exact sequence of a Kan fibration of sheaves of simplicial sets and Barr's results were exactly what I needed. All this has since been improved by using closed model structures and a less naive definition of fibration, but the "embedding/technology" Ross mentions was all we had available--in sufficient generality to cover my situation--at the time. Don Van Osdol
Would the following (aside from the first) help to prepare a more digestible proof of Barr's full regular embedding theorem? 1) Say that an object U of a (but not just any) category E is open if U is a finitely presentable object of the opposite of E (i.e., if it satisfies the analogue of the filter convergence characterization of open sets). An object P of E is strict if it is the projective limit of a diagram of opens such that each limit projection is an effective epimorphism; P is called saturated if for every U -->> V of opens, each P -->> V lifts to some P -->> U; finally, P is called weakly projective if it is projective with respect to effective epimorphisms of opens. Every strict, saturated object is weakly projective. 2) Given P in Pro(C), where C is a regular category, let P' denote the fibred product of all X -->> P which are opens of Pro(C)/P, and P* the projective limit of: P <<-- P' <<-- P'' <<-- P''' ...; one shows that the projection P* --> P is an effective epimoprhism, that P* is a strict, saturated object of Pro(C)/P, and therefore that P* is a weakly projective object of Pro(C). Theorem 14 of Barr's '86 JPAA paper (i.e., step three in his message of 21 May) applies. 3) The construction is (to my knowledge) due to Joyal (c. 1972, or earlier). Objects of the form X*, where X is an object of C, are principal prime models, a notion Makkai (Full continuous embeddings of toposes, Trans AMS, 1982) distilled from Barr's original proof (Joyal called them absolutely flasque). cheers, -b PS:Without the terminology, that Pro(C) (=~ Lex(C,Set)^op) is regular if C is is suggested in SGA4, Expose I, 8.9 (especially cf. exercise 8.9.9(b)).
Well, I am not trying to be snide, but for me, my argument is more understandable. But if you prefer to see it in the way you put it, I have no objection. A couple more comments. In 1986 (or, really, 1984), I did not really understand that this argument was *really* the same argument, just dressed up nicer, using functors instead of diagrams. In 1970, I was fresh from homological algebra. The construction struck me (and still does) as reminiscent of the argument you use in showing that if you have the beginnings of a map from a projective complex to an acyclic one, you can continue it one more step. The main difference was that this was a well-founded poset instead of an inter-indexed chain. And if anyone ever wondered why I called these A diagrams and P diagrams, it had nothing to do with A&P and everything to do with acyclic and projective. Thus the embedding theorem is, like many other things I have done, a form of acyclic models.
participants (4)
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Donovan Van Osdol -
Michael Barr -
Ockham's stubble -
street＠mpce.mq.edu.au