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nLab > Latest Changes: reflexive coequalizer

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeDec 25th 2009
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012
    • (edited Nov 26th 2012)

    Somebody asks me by email where the proofs for the claims at reflexive coequalizers can be found. Apparently not all of them are in the single reference cited there, and the question is to who to attribute them. But I haven’t checked, no time right now. Somebody who feels responsible for this please check. (I suggested to that person to forward his question to here, but I don’t know if he will.)

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 26th 2012
    • (edited Nov 26th 2012)

    Urs, I think there are other entries which indicate the proofs, but I’m happy to give them here as well (at least some of them). Linton was of course cited.

    Edit: I recall that a proof of proposition 1 is given on page 1 of Johnstone’s Topos Theory (baby elephant), at least in essence. But somebody who has that book at hand can hopefully confirm.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012

    I understand the question as mainly asking for who to attribute these statements to, as some of them are apparently not in Linton. I haven’t checked. But if you can add a remark clarifying this, it would be great in any case.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 26th 2012

    Well, I’ve added a proof of proposition 1 and some other things, and I cited things as best as I have direct knowledge of. Some of this might be folkloric; I’d maybe take a peek at the Johnstone reference, but I don’t think he cited a source for the lemma on page 1. Some of this reprises material in colimits in categories of algebras.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012

    Thanks, Todd! That’s awesome. I’ll notify my correspondent of these additions (in case he didn’t follow my pointer to have a look here at the forum…)

  7. Link was dead

    Anonymous

    diff, v11, current

    • CommentRowNumber8.
    • CommentAuthorjonsterling
    • CommentTimeMay 20th 2023

    I don’t think the description of Linton’s proof is correct, as it uses the algebra map for A iA_i nowhere, but it should appear in part of the reflexive pair. Am I missing something?

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 20th 2023

    I think you’re missing the fact that the algebra map for A iA_i is baked into the counit component εA i\varepsilon A_i.

    • CommentRowNumber10.
    • CommentAuthorjonsterling
    • CommentTimeMay 20th 2023

    Ahh, thanks Todd, that must be it! Incidentally, aside from this non-issue, the colimit computed here is still seems different from the one used by Linton in Corollary 2 of “Coequalizers in categories of algebras”. But I guess there are many roads to Rome :)

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 20th 2023

    I don’t have easy access to that article from where I sit. It’s curious that it’s different, though, since the nLab proof seems as natural as falling off a log. :-)

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023
    • (edited May 20th 2023)

    I have added the DOI: doi:10.1007/BFb0083082.

    With the DOI in hand, you get the article from sci-hub: here it is.

    diff, v14, current

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 20th 2023

    Thanks, Urs! I didn’t know this neat trick.

1 to 13 of 13

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