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nLab > Latest Changes: topos of algebras over a monad

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2011
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeMay 18th 2011

    I added a paragraph in the literature, justifying and having a link to vectoid.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2011

    Thanks, that in fact gives me finally a helpful way to think about vectoids. I didn’t appreciate this point before.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2011

    I have added the statement that TCoAlg(ℰ)T CoAlg(\mathcal{E}) is a sheaf topos if ℰ\mathcal{E} is and TT is not only lex but also accessible. I guess I can see this (will try to spell out all the proves for the entry eventually). But where is this discussed in the literature?

    • CommentRowNumber5.
    • CommentAuthorRodMcGuire
    • CommentTimeMay 18th 2011

    Nikolai Durov’s Classifying Vectoids and Generalisations of Operads has just (May 16, 2011) appeared in arxiv.

    So I guess the Russian can be taken out of the algebrad page.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeMay 18th 2011
    • (edited May 18th 2011)

    Thank you very much Rod.

    But what do you mean Russian version “taken out” ?? The Russian version

    • “Классифицирующие вектоиды и классы операд”, Trudy MIAN, vol. 273

    is the published version and it is the original. The original should never be removed from the record.

    P.S. I did not put before the Russian version of the paper though to the page (now I did). I have the copy for a year or so, but I posted just the conference part as the paper did not appear publicly at the time. I now added the publication data. Conference digest could stay as well.

    I have now also added the reference to Nikolai Durov.

    • CommentRowNumber7.
    • CommentAuthorRodMcGuire
    • CommentTimeMay 18th 2011
    • (edited May 18th 2011)

    By “Russian taken out” I was just referring to the Russian text abstract included in the entry.

    I was also wondering if the arxiv version is not merely a translation of the Trudy MIAN vol 273 version but also included new material since you say it is a year or so old, and no date is given for its publication.

    However I just checked and Vol 272 is the latest issue so in some since the Russian version hasn’t been published yet.

    Is there a difference between the two versions?

    [ UGH. I don’t seem able to have an ’a’ tag include ’option_lang=eng’ for ’Vol 272’ above ]

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeMay 18th 2011

    The Russian abstract is the abstract of the talk, not the paper. The paper has been written a year ago, accepted some time ago and got the issue number which is yet not printed, but the issue belongs to this year. The English version which I suppose hasn’t got much difference from the Russian version will be published next year, in the English edition.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 9th 2013
    • (edited May 9th 2013)

    Urs wrote (in #4):

    But where is this discussed in the literature?

    I am also wondering this. The Elephant only gives a passing mention.

    EDIT: only another passing mention, but enough to construct a proof I think. Richard Garner’s paper on ionads mentions that the category of coalgebras for a lex comonad is locally presentable (hence a Grothendieck topos) iff the comonad is accessible.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 9th 2013

    I performed some edits and added some material at topos of algebras over a monad.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 9th 2013

    I’d like to propose that this page be renamed to topos of coalgebras over a comonad, which is what it’s mostly about. The material on algebras is adequately covered in other pages that are referred to.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2015
    • (edited Aug 28th 2015)

    Given an accesible left exact comonad on a sheaf topos eqipped with a site of definition, I’d like to have an explicit site of definition of the topos of coalgebras.

    Specifically I’d like to have this for the jet comonad acting on a slice of the Cahiers topos over some smooth manifold Σ\Sigma.

    Conversely, I suppose there is an evident structure of a site on the category of smooth bundles over Σ\Sigma with (nonlinear) differential operators between them as morphisms, hence on the restriction of the coKleisli category of the jet comonad to bundles of manifolds. How might the sheaf topos on that site be related to the topos of algebras over the comonad acting on the Cahiers topos?

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 28th 2015

    That’s a very good question. I also think it would be interesting to contemplate what theory coalgebras classify (a question which could stand some refinement).

    • CommentRowNumber14.
    • CommentAuthorZhen Lin
    • CommentTimeAug 28th 2015

    Getting generators for the category of coalgebras for an accessible comonad is a lot less obvious than the category of algebras for an accessible monad. Perhaps if someone knew an explicit argument for showing that the category of coalgebras is accessible…

    • CommentRowNumber15.
    • CommentAuthorspitters
    • CommentTimeAug 28th 2015

    Johnstone - “collapsed toposes” and “when is a variety a topos?” and “cartesian monads on toposes.”

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeAug 29th 2015

    Well, one could presumably beta-reduce Makkai-Pare’s limit theorem for the case of coalgebras..

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2015
    • (edited Aug 29th 2015)

    Thanks for the replies.

    Bas, thanks for the references, but does any one of them address the question that I am asking: how to get a site of definiton for the topos of coalgebras of a left exact accessible comonad acting on a topos?

    By the discussion at topos of coalgebras over a comonad I already know that my category of coalgebras is a topos. What I am after is a site of definition for it.

    I don’t need this site generally, just for the case of the jet comonad acting on the slice of the Cahiers topos over any manifold Σ\Sigma.

    Or while I am at it, what I really-really need is to relate the topos Sh(DiffOp Σ(Mfd))Sh(DiffOp_\Sigma(Mfd)) of sheaves on the site DiffOp Σ(Mfd)DiffOp_\Sigma(Mfd) (which is the restriction of the full coKleisli category to those objects in the Cahiers topos represented by Frechet manifolds) to the full topos of all coalgebras.

    Now I don’t expect that to be helpful information to anyone, but that’s what I am really after.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2015

    Ah, I am being stupid. For that special case apply comonadic descent and it should follow that a site of definition is the comma category of the inclusion of the original site over ℑΣ\Im \Sigma.

    • CommentRowNumber19.
    • CommentAuthorspitters
    • CommentTimeAug 29th 2015

    Urs, I did not mean to imply that Johnstone’s papers answer your question, just that they are considered with the question when do T-algebras form a topos. I’ve added the references to the page for posterity.

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