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nLab > Latest Changes: images preserve unions

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2017
    • (edited May 19th 2017)

    in the course of writing out proofs in elementary topology, i found it useful to be able to easily link to elementary statements in elementary set theory. So I created entries like

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 19th 2017

    This being the nLab, I would strongly urge that some explanation in terms of adjoints (specifically, an adjoint string ∃ f⊣f *⊣∀ f\exists_f \dashv f^\ast \dashv \forall_f) be added. I would guess that most mathematicians do not know these properties in their bones, and a good time to start hammering this type of thing home is right in the undergraduate classroom.

    I will probably add something along these lines soon, if there is no objection.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2017

    if there is no objection.

    Certainly not!

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 19th 2017

    I wrote in, rather quickly, some arguments and remarks on this. But it may need more polishing. (More later, but I must dash now.)

    The statement about countable unions of countable sets requires the countable axiom of choice, it should be noted.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2017
    • (edited May 19th 2017)

    Thanks, Todd. I have added a remark here mentioning yet more jargon, such as “base change” and “hyperdoctrine”. This could be much expanded further, but I also need to be doing something else now.

    Eventually we want this kind of general discussiono at a more central place that we could link to from other entries, too. Maybe an entry titled adjunctions in elementary set theory or the like

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 19th 2017

    Urs, I made a few adjustments which I hope you don’t mind. It seems to me that thematically it works better to unify the pages on how images and pre-images interact with unions and intersections, so I renamed the page. To the Statements section I added the proposition on inverse images preserving unions and intersections, and I made a slight correction of one of the statements about direct images along injective functions. I haven’t harmonized all the notation the two of us are using.

    There is at least one more thing I might want to add to your “jargon-y” remark, on Frobenius reciprocity, which relates to the bit about injective functions.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2017
    • (edited May 20th 2017)

    Thanks, Todd, that’s beautiful!

    In that “jargon”-remark I added pointer to the general fact that adjoints preserve (co-)limits and then I used the occasion to give that statement a page of its own, too.

    I have this vision now that we might eventually augment the series

    with an “Introduction to Set Theory” that would proceed along just these lines: Give an introduction to traditional elementary set theory for readers with no background, but in doing so provide, without much ado, the correct picture that will make the reader inadvertently learn the categorical logic and type theory.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMay 20th 2017

    This is a very nice-looking page, thanks guys.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2017

    I created a stub entry for Mikhail Yakovlevich Suslin, in particular since the first hit for “Suslin” is Andrei Suslin.

    Todd, where it says in the footnote “There is a famous story…”, would it be easy to add some citation for this? Just for completeness.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2017

    While we are at it, I also gave an entry to hom-functor preserves limits.

    But now I really need to be doing something else.

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 21st 2017

    Thanks for the nice feedback! I’ll add a citation for the story.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 21st 2017
    • (edited May 21st 2017)

    Re #7: I like that vision, and I agree with the “not much ado”. I firmly believe it’s possible to teach undergraduates this way.

    IIRC, somewhere in Paul Taylor’s book he likens the gentle introduction of categorical thinking just at the level of posets to a “trivium” (as in classical education), and the more concerted study for more general categories to a “quadrivium” – something at a higher level for those who have passed initiation through the trivium. Pedagogically, I think that’s exactly right, and when giving undergraduates their first taste of modern mathematics, e.g., their first real analysis course, you can sneak in categorical thinking almost entirely at the trivium level, where it gets used a lot (properties of sups and infs of sets of real numbers, unions, intersections, direct and inverse images, etc.). You don’t even have to say the words “category theory”, you just do it like it’s the most natural thing in the world (which it is).

    • CommentRowNumber13.
    • CommentAuthorRodMcGuire
    • CommentTimeMay 22nd 2017
    • (edited May 22nd 2017)

    [edit] ugh. It posted before I had finished. I’ll try to finish tomorrow.

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 23rd 2017

    Referring back to #1 in this discussion, I’ve added the proof (or proofs, both classical and constructive) to countable unions of countable sets are countable.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2017

    Thanks!

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