I'm going on memory here, but I may have picked up on the ideas in the first paragraph by reading the first few chapters of Handbook of Mathematical Logic. Johnstone's book is also really good, and goes into a lot of detail. We're adding bits and pieces to the nLab (e.g., stuff surrounding "ultrafilter theorem"). As for the stuff in the last paragraph, you probably have Mac Lane-Moerdijk as a reference; the stuff from Lawvere comes from his Chicago lecture notes Variable Sets, Etendu, and Variable Structures in Topoi. Sorry I don't have more!

Not the first time the question has been asked. See "Is the presheaf category of a locally small category locally small?"

Eric - sorry. If your local university library doesn't have the La Jolla Conference Proceedings, and if you can't get it easily through interlibrary loan, then you might try accessing through gigapedia.com (which I have only just found out about myself). I'm not sure I have a photocopy that I can email you. As for Kreisel, there are several articles; try the appendix to Feferman's article in Springer LNM 106, and also check these slides: math.mcgill.ca/rags/seminar/Marquis_KreiselLawvere.pdf. In some sense Kreisel's is a "party line" against "categorical foundations". HTH!

Hi Eric. The only source I know of off-hand for this is something you may already know: Lawvere's article The Category of Categories as a Foundation for Mathematics, in Proc. Conf. Categorical Algebra – La Jolla., Eilenberg, S. et al., eds. Springer-Verlag: Berlin, Heidelberg and New York., pp. 1–20. And yes, John Isbell found some problems with it. But certainly the idea of the article was in a structuralist mode. I'd also like to mention that I think different people have different ideas about what "foundations" should mean, and there is a oft-cited article by Kreisel on this very question.

To be more precise, Harry, ETCS and SEAR are first-order theories, just as ZFC is. ZFC is a one-sorted theory with a binary predicate, and ETCS can be made a one-sorted <i>essentially algebraic</i> theory (although two sorts, "object" and "morphism", are more common here). Now you slip in what could be weasel words, "informal notion of class", but how is the situation with ZFC any different "informally"? Koichi: that was my point: what those members "are" is irrelevant to most practicing mathematicians.

Obviously there's a lot to add, but you're right: these little comment boxes are not the best place. There are various possibilities for "alternative foundations" which avoids these strange glitches. One is ETCS (due to Lawvere), which you can begin reading about at the nLab or from <a href="tac.mta.ca/tac/reprints/articles/11/… himself</a>. (But for some people, ETCS is hard and technical, and not as intuitive as ZFC.) Another very intriguing possibility is Mike Shulman's SEAR (exclusively at the nLab, AFAIK).

Yes, under the sorts of set-theoretic encodings you appear to have in mind, the argument looks to me impeccable but still "morally wrong"! (It would mean $\hom(F, G)$ is not small even if $G$ is terminal.) It's an excellent illustration of the kind of sophistry that's possible by encoding everything as sets and sets as membership trees. More satisfactory would be a foundations where this type of argument cannot even be formulated (cf. discussion of structuralist vs. materialist forms of set theory in the nLab). Suffice it to say that the problem is not insurmountable. :-)

I meant not with regard to your counter-example, but with regard to your plea at the end (but maybe you understood that's what I meant! :-)

Andrej, did you take a look at the Freyd-Street paper? Because their construction is quite explicit and concrete: see their definition of a functor "T".

Even if $C$ is large, the collection of transformations $F \to G$ between presheaves can be essentially small (in definable isomorphism with a set). Consider for example $F$ representable and apply Yoneda.

I am not absolutely sure, but I suspect the isomorphism/equivalence distinction for categories must have taken some time to really seep into the general collective consciousness (and it's still taking place). I have to think that Mac Lane's book was very state-of-the-art back in the early 1970's, and it would be interesting to know who if anyone was complaining back then about what seems obvious to us now. In fact, having grown up on CWM, I can sort of recall my own first encounter with "creates" vs. "reflects", and thinking then it was something very subtle, almost pedantic! :-)

That's one possibility, and I hope that's the explanation. I can think of another explanation in the case of Mac Lane: he could be very stubborn, and had a famously strong temper (Ieke has spoken publicly about the heat of some of their writing sessions, adding that with Mac Lane you could always start again fresh the next day, but maybe he didn't care to fight this one out). It's a beautiful book, though, written with a lot of consideration for the reader, and knowing both authors, one can feel the very collaborative nature of it.

Update: I went ahead and contacted Michel Coste about this, and the answer is apparently 'yes' according to some work by Jes&uacute;s Escribano which I now have. I don't know if this means the discussion should now be closed, but I'm happy to keep it open in case anyone else has something illuminating to say.