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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeSep 29th 2010

    Discussion about historical influence of MacLane’s CWM textbook from a query box in nlab is transferred here. Toby, Todd or Mike, if you want it back please rollback the entry. I put the link to the original entry.

    The reaction is to

    Categories Work is the standard reference for category theory, and we may often cite it here. Almost all of its terminology is widely adopted, although strangely its approach to foundations is not.


    Todd asks: In what sense do you mean “strangely”? Do you mean that it’s strange that more categorists don’t like Mac Lane’s approach because it deserves better, or that it’s strange that Mac Lane missed here when he so often got things right? Or something else?

    Toby replies: I mean something more neutral: It's strange that people don't follow him here, when they follow him on so much else.

    Mike Shulman: I’m curious why you say that people don’t follow him. Mac Lane’s approach to foundations is, I believe, the assumption of one Grothendieck universe. Most non-category-theorists who I talk to, and many category theorists, don’t really make any explicit foundational choice (perhaps partly because of ignorance of the options); thus I don’t think they could be said either to follow or to not follow any particular approach. And of the category theory I’ve read which does make reference to foundations, one universe seems to be a fairly common assumption.

    Toby: Maybe I'm thinking of less sophisticated references than you are, but it seems to me that when people just dash off a definition of category, trying to make sure that it's correct but otherwise not giving much thought to foundations, that they almost invariably adopt the set/class distinction of NGB set theory. And this is equivalent to ZFC, weaker than ZFC plus one uncountable Grothendieck universe. Someone taking more care with foundations might well prefer to assume an uncountable Grothendieck universe (or equivalently an uncountable inaccessible cardinal), which makes things easier to work with, but I don't see it otherwise. (I suppose that now we should hunt down specific references to make our respective points.)

    Mike: I don’t really think I ought to spend the time hunting up (or down) references on such a not-very-consequential point. Now that I think about it, you’re right that many people do think of large categories as proper classes. But I think there are also enough of people who like to think of at least one universe (and usually need no more than one). Maybe we could leave it at “some do, some don’t”?

    Toby: I'll agree to that, although I stand by the ’not’ that modifies ’widely adopted’ in this case. If you think that you know how you'd like to phrase it, please go ahead.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeSep 30th 2010

    Thank you.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeSep 30th 2010

    Link to article: Categories Work.

    Looks fine to me.