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nLab > Latest Changes: monadicity theorem

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2010

    at monadicity theorem in the second formulation of the theorem, item 3, it said

    CC has

    I think it must be

    DD has

    and have changed it accordingly. But have a look.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2010

    Just for definiteness I stated (again) at conservative functor the property that such reflects all (co)limits wich it preserves

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 17th 2010

    Yes, that seems right to me.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeNov 17th 2010

    Then this would be true in particular for embeddings (fully faithful functors) of categories. If there are no limits of some type within original subcategory, then they are by definition preserved; on the other hand such limits are not reflected if they do exist in the target ambient category. Where is the error ?

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 17th 2010

    In other words, the proposition says that if limKlim K exists in CC, and if U:C→DU: C \to D is a conservative functor such that U(limK)U(lim K) is the limit of UKU K in DD, and if const c→Kconst_c \to K is a cone for which the induced cone Uc→UKU c \to U K is the limit, then cc is the limit of KK. So one of the hypotheses is that the limit exists in the subcategory.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeNov 17th 2010

    Thanks for more detailed/precise statement, Todd.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeDec 6th 2010

    I added a couple more examples to monadicity theorem.

  8. Added a missing condition in the ‘specifically this means’ description of split coequalizers – the fact that the arrows form a fork doesn’t follow from the other conditions.

    Jonas Frey

    diff, v30, current

    • CommentRowNumber9.
    • CommentAuthorJohn Baez
    • CommentTimeFeb 16th 2021

    Added a bit more info on crude monadicity.

    diff, v33, current

    • CommentRowNumber10.
    • CommentAuthorvarkor
    • CommentTimeSep 16th 2021

    Add a scan of the “untitled manuscript” of Jon Beck containing the original proofs of the monadicity theorems, provided by John Kennison.

    diff, v34, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2021

    Thanks! I have adjusted the formatting of the bibitem a little (here) and copied it over also to the entry on Beck (here).

    diff, v35, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 11th 2022

    added an elementary example (base change of presheaves along an essentially surjective functor): here

    diff, v39, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2022

    added also the example (here) of the forgetful functor on group actions

    diff, v40, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2022

    added also mentioning of the case of monoid actions: here

    diff, v41, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2022

    While I am at it, I have polished up formatting and hyperlinking of the old Examples-section “Groups over sets” (here)

    diff, v41, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2022

    just for completeness, I gave the entry a sentence under “Idea”.

    diff, v43, current

  17. Corrected section number in the reference to Lurie HA.

    Mark Barbone

    diff, v49, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeAug 2nd 2023

    I have expanded out the third clause in the “alternative formulation” (here) in order to dispell ambiguity.

    diff, v50, current

    • CommentRowNumber19.
    • CommentAuthorvarkor
    • CommentTimeMay 7th 2024

    Added some references.

    diff, v54, current

    • CommentRowNumber20.
    • CommentAuthorvarkor
    • CommentTimeJun 21st 2024

    Added a reference to another monadicity theorem due to Borceux and Day.

    diff, v55, current

    • CommentRowNumber21.
    • CommentAuthorvarkor
    • CommentTimeJun 21st 2024

    Added two more relevant references of Linton.

    diff, v55, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2024

    added DOI-s and other missing publication data,

    and copied these items also to their respective author pages

    diff, v56, current

  23. Doesn’t seem like anyone can justify this sentence, removing.

    kevindcarlson

    diff, v58, current

1 to 23 of 23

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