# How different category theories relate

Mac Lane "Categories for the Working Mathematician" and "Abstract and Concrete Categories. The Joy of Cats" use different set theory foundations.

How one to transfer theorems between these two different systems? That is if a theorem is proved in one of these two systems what can be inferred in the other?

-

Adrian Mathias has written some excellent articles comparing the specific set theory used by Mac Lane and used in other parts of category theory.

• His article The strength of Mac Lane set theory is a detailed analysis of the strength of Mac Lane's set theory in comparison with other theories.

• Adrian Mathias, "What is Mac Lane missing?" (a comment on the foundational stance of Saunders Mac Lane; MR 94g:03010; published with a reply by Mac Lane, MR 94g:03011, in Set Theory of the Continuum, ed. H. Judah, W.Just, H.Woodin; Mathematical Sciences Research Institute Publications Volume 26, Springer-Verlag, 1992.)

• Further papers available on Mathias's web page depository.

-
Are you supposing me to read and understand all this? I'm not a logician. But my quick search for the word "universe" in "The strength of Mac Lane set theory" reveals that that article does not even mention Grothendieck universes. How is this related with the logical system of "Categories for the Working Mathematician"? –  porton Jun 17 '11 at 20:35
Oh, if you don't find my answer helpful, then kindly please ignore it. Meanwhile, Mathias' work is quite interesting, although it is true that the systems he analyzes in the first article seem not to involve universes directly. The usual arguments I have seen giving the strength of the axiom of universes as inaccessible cardinals presupposes ZFC as a background theory, and if you weaken it to the theories Mathias attributes to Mac Lane, then I'm not sure how that comparison is affected. –  Joel David Hamkins Jun 17 '11 at 20:48
@porton: your "Are you supposing me to read and understand all this?" is probably the worst you can say to someone who took the time to write such a helpful answer... –  Mariano Suárez-Alvarez Jun 17 '11 at 21:10
So-called "Mac Lane set theory" is not however the same as the set of assumptions formally adopted in the section on Foundations in Categories for the Working Mathematician. Mac Lane set theory is a membership-based set theory equal in strength to Lawvere's Elementary Theory of the Category of Sets (ETCS); it's not quite as strong as the assumption of ZFC + one inaccessible adopted in CWM. –  Todd Trimble Jun 18 '11 at 0:04
"not quite as strong" is quite an understatement! –  Mike Shulman Jun 18 '11 at 5:36
show 3 more comments

I haven't looked super-carefully at the assumptions in The Joy of Cats; they are described in a slightly hand-wavy way (the reader is referred to the appendix in Herrlich-Strecker, which I do not have to hand). But it's pretty clear that an assumption of ZFC plus two strong inaccessibles, one containing the other, is more than sufficient for the purposes of The Joy of Cats (they have sets contained in classes, and classes contained in "conglomerates", and they have some set-theoretical assumptions on conglomerates, the most serious of which is that a product of conglomerates indexed over a conglomerate is a conglomerate).

The formal foundations in Categories for the Working Mathematician suppose: ZFC + one inaccessible.

From the standpoint of a professional set-theorist, I think either set of assumptions would be considered fairly mild (at least when put up against large cardinal hypotheses at which a set theorist would not bat an eye), and the reaction of most people would be not to worry too much about the difference. Without having gone thoroughly through The Joy of Cats, I should think that any theorem therein that does not mention the word "conglomerate" (which might be on occasion tacit but not difficult to detect, as in "the category of categories of at most class size") would be a formal theorem under Mac Lane's declared foundations, and I am also pretty sure that Mac Lane (whom I got to know) would have no difficulty accepting ZFC + two strong inaccessibles to deal with the remainder -- it's just that he didn't need that assumption to write his book.

Without having a more specific focused question to deal with, I'm not sure one can make a more positive, guaranteed-to-be-true blanket statement about a text which is several hundred pages long.

-
But "Joy of Cats"'s "inaccessibles" are not sets, one of them is a proper class and the other is a conglomerate. Doesn't it affect anything? –  porton Jun 18 '11 at 11:18
Let's not get too hung up on words. You could call the sets of rank below the first inaccessible "small sets", those of rank below the second inaccessible "moderate sets", and those of arbitrary rank just "sets". Or, if you like you could rename them "sets", "classes", and "conglomerates". My point is that whatever you call them, "ZFC + two (strong) inaccessibles" more than suffices as a formal foundation for The Joy of Cats. –  Todd Trimble Jun 18 '11 at 13:04
@Mike: I saw your paper when it appeared, and benefited from reading it. As I recall, in your paper, there was no reference to Solomon Feferman's work on the foundations of category theory as it relates to the Quine-Jensen set theory $NFU$, e.g., his paper Enriched Stratified Systems for the Foundations of Category Theory, that appeared in the 2006 Collection What is Category Theory [the paper is also available on Feferman's homepage]. Was that intentional? –  Ali Enayat Jun 18 '11 at 18:14