Timeline for Why need the morphisms to form a set ?
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| Dec 14, 2010 at 1:20 | comment | added | Ralph | Ryan, thanks for the reply. I'm neither a set theorist and don't know under which restrictions one can build a "category of classes". In any case I agree that in practise Yondea's lemma is most reasonable in conjunction with (locally) small categories. | |
| Dec 12, 2010 at 19:59 | comment | added | Ryan Reich | @Ralph: also, in the comments mentioning the category of all categories, I think they mean small categories. Look at the third answer to mathoverflow.net/questions/3278/… for a discussion of this issue. | |
| Dec 12, 2010 at 19:59 | comment | added | Ryan Reich | @Ralph: It depends what you mean by "class". I am not a set theorist, and I know of only two theories which treat them: NBG (von Neumann-Bernays-Goedel) and Grothendieck universes. Neither one allows a class to contain itself. The ultimate reason is that, without some form of restriction on the axiom of comprehension, doing so would allow Russell's paradox. In that Feferman paper, it appears (though I don't know because, not being a set theorist, I didn't read it carefully) that he is using a set theory with such a restriction, based on types. | |
| Dec 10, 2010 at 16:07 | comment | added | Ralph | Why is there "no such thing as the class of all classes" ? In some comments above the category of all categories is mentioned. So naively it's not hard to consider a category with objects all classes and morphism all mappings between two classes. And in fact this construct appears in 6 iv) (page 10) of the article: math.stanford.edu/~feferman/papers/ess.pdf There is also mentioned a Yoneda lemma for arbitrary (i.e. not necessarily locally small) categories. | |
| Dec 9, 2010 at 19:50 | comment | added | Adam Hughes | I quite agree. Yoneda is certainly the most useful tool in category, at least in all the things I care about. | |
| Dec 9, 2010 at 19:47 | history | answered | Ryan Reich | CC BY-SA 2.5 |