How different category theories relate - MathOverflow

How different category theories relate

2011-06-17T19:09:29Z

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?

Answer by Joel David Hamkins for How different category theories relate

2011-06-17T20:24:53Z

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.)
See also Adrian Mathias, "Strong statements of analysis."
Further papers available on Mathias's web page depository.

Answer by Todd Trimble for How different category theories relate

2011-06-18T00:27:01Z

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.

Answer by Mike Shulman for How different category theories relate

2011-06-18T05:39:28Z

You might be interested in this paper (although it is in need of revising).