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Created General Theory of Natural Equivalences, partly on the model of the (to me) useful-seeming Elephant. One reason was that I think it can be instructive for people learning more category theory to see this classic transparently commented on in a modern reference work like the nLab.
Another reason was that this is a historical important paper, which I needed an nLab reference for when writing a historical, context-adding section of directed graph, embedding Lawvere’s interesting Como comments into a relevant context, drawing historical parallels.
Peter, what is an email address by which you may be reached?
Urs, my last name at ma.tum.de
It only applies to small categories, since otherwise the definition “$R(A) =$ the set of all morphisms with codomain $A$” need not be a (small) set.
No, I don’t think it is at all controversial, and it is inarguably mathematical. There are many ways to formalize the distinction between “small” and “large” (I wrote this note about a bunch of them), but it’s hard to dispute that some such distinction is necessary. In particular, the category of all (small) sets cannot be a small category, on pain of Russellian paradoxes.
Hi, I find your point of view interesting! Thanks for elaborating on it and for asking your question here!
I think if it were purely a matter of Russellian paradoxes, then I would be very sympathetic to your point of view. But there are situations in category theory where size seems fundamental. One example might be the fact that any category with all colimits (not only small ones) is a poset. Perhaps this theorem is also artificial in some sense, but it would be interesting to explore in what way it is so.
Note that from a structural point of view, even the material classes of ZFC are extremely well-behaved. You can do almost everything with them that you would do with sets, as they form a well-pointed, Boolean infinitary pretopos with subobject classifier. All one lacks for this to act the same as the category of sets is Cartesian closedness and a strong version of Choice (surjective class functions have sections). I view this as at least a partial validation of any Platonist view about any manipulations of classes as required by mathematics. Mike knows, I suspect, a more precise alignment of this informal view with the conservativity of NBG over ZFC, and the relation of all this to Algebraic Set Theory.
Once has the category C of classes (that is, something obeying the first-order category axioms), large categories, in the sense the total collection of morphisms is a class, become internal categories in C, hence essentially algebraic objects.
What seems missing, though, is an expositon of these facts which would make them accessible for the masses.
outside of questions that has clear set theoretical character
It must be admitted that such questions cut a huge swath. Category theory is permeated by smallness considerations, as is obvious from the adjoint functor theorem, Giraud’s characterization of Grothendieck toposes, the theory of accessible and locally presentable categories, the small object argument – the list goes on and on.
This is not to say the final word has been uttered on the small-large division – all sorts of ideas should be entertained. I am disinclined to believe however that it’s really a question of “maturity”: limitations of one kind or another are simply baked into mathematical reality.
@Urs, do you mean #10? If so, I agree. It’s hard to educate people :-) My only remaining internal quibble is how to properly say ’classes form a category’ in a way that will minimise people arguing about how one should think about collecting all classes into a single collection. The moment you say ’model of a first-order theory’, people tend to think a model is a set. Maybe it’s down to ZFC classes being somewhat syntactic.
Yeah, if pressed I’d personally go with the option of dealing with syntactic objects in a meta-theory (which we can treat or pretend to treat “Platonistically”, i.e., taking infinite collections of syntactic strings and their manipulations as God-given). We’ve made little starts here and there already I think, especially when we were discussing Scott’s trick not too long ago.
I’d also drop this paper here as an interesting approach:
Paul Blain Levy, Formulating Categorical Concepts using Classes, arXiv:1801.08528
Apologies for not responding to #8 sooner; I’ve been busy with the semester starting, and it’s been a bit tricky to figure out what I want to say.
I definitely still agree, as I wrote in that note and as Richard said in #9, that there are situations in modern category theory where size feels fundamental. Some of them are much more naturally-occurrring too, for instance the requirement of accessibility on functors and categories for free algebras to exist.
However, I have for some time now also entertained privately the belief that there should also be some kind of “unlimited category theory” (a phrase coined by Feferman) that is “free of size”. If such a theory existed, it would necessarily appear bizarre in some ways to a modern category theorist, since it would have to avoid the paradoxes of size somehow. (In particular, Feferman’s desiderata for such a theory appear to be contradictory, although there’s a bit of wiggle room for debate since he formulated them imprecisely.) But in other ways, it could feel like a dream come true. (No more checking of solution-set conditions and accessibility? Sign me up!) In particular, I share your intuition that $Set$ “really” means the category of all sets, and $Cat$ the category of all categories, including itself — as long as you don’t do anything stupid with it. If nothing else, it would be interesting to explore such a theory if it existed, even if it ended up being too weird to supplant ordinary size-based category theory for most applications.
In fact there is no shortage of fully rigorous proposals for “size-free” foundations of mathematics, including stratified set theories like NF, “positive” set theories, and also set theories based on linear logic and paraconsistent logic. I don’t find any of them really satisfactory, but that could be because no one has taken the time to seriously work out what category theory would look like therein. It would be a worthy project! (I actually also have my own idea for how to obtain such a theory, but I’m not ready to talk about it publically as many details remain to be worked out.)
Perhaps the best answer to the question you asked of “what should my attitude be” is that I think a better analogy to the history of calculus is to consider the current situation, with size distinctions formalized using proper classes or Grothendieck universes, as corresponding to the $\epsilon$-$\delta$ formalism. Both are a successful way to give a fully rigorous grounding to a subject, yet nevertheless still fail at capturing something of the original intuition. The still-hypothetical unlimited category theory would then correspond instead to the rigorous infinitesimals of Robinson’s Nonstandard Analysis or of Synthetic Differential Geometry, which don’t behave exactly like the unrigorous ones of the early pioneers, but are arguably closer to what they had in mind than $\forall \epsilon\exists \delta$.
If I am reading MacLane right, he calls the Grand Set Theoretic Doctrine
Mathematics is what can be done within axiomatic set theory using classical predicate logic.
I think this is unjustifiable as a “definition” of mathematics, if possible even more so today than it was when MacLane wrote it. However, that doesn’t entail a total rejection of “foundational theories”. As I’ve written elsewhere, the distinguishing feature of set theory is not that all of mathematics is done inside of set theory, but that all (or at least much) of mathematics could be “coded up” inside of set theory. This is analogous to how a Turing-complete programming language can simulate any other language, or an NP-complete problem can have any other NP problem reduced to it; so a more neutral term than “foundational” would be something like “mathematics-complete”. There’s a lot of value in studying such theories and the ways in which they can represent all of mathematics, as long as we don’t put the cart before the horse and start identifying mathematics with its encoding into such a theory.
Added:
Transactions of the American Mathematical Society 58 (1945), 231–294. doi:10.1090/S0002-9947-1945-0013131-6.
The first page says
Presented to the Society, September 8, 1942; received by the editors May 15, 1945.
completed the bib-item, added link to jstor:1990284 and cross-linked with category and category theory (where the item appeared unlinked to this page here)
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