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    • CommentRowNumber1.
    • CommentAuthorThomas Holder
    • CommentTimeOct 21st 2018

    Expanded and reorganised the entry.

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 21st 2018

    Does the dot FGF \cdot G between two functors indicate composition in traditional Leibnizian order,or does it mean “FF followed by GG”? Because if the former, I think the universal arrow ought to be instead α:(Ran GG)GG\alpha: (Ran_G G) \cdot G \Rightarrow G.

    • CommentRowNumber3.
    • CommentAuthorThomas Holder
    • CommentTimeOct 21st 2018

    Thanks for catching this!

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 29th 2019

    Added Avery’s result about the Giry monad as an example.

    It’s not a terribly friendly Idea section. ’Condense functor’ redirects to dense functor and from there one can reach dense subspace. But we have a page codense functor, so I’ll make it point there.

    diff, v6, current

    • CommentRowNumber5.
    • CommentAuthorThomas Holder
    • CommentTimeJan 29th 2019

    Reworded the idea section.

    diff, v8, current

    • CommentRowNumber6.
    • CommentAuthorSam Staton
    • CommentTimeAug 16th 2019

    added description of codensity in terms of adjunction with presheaf category, from Tom Leinster’s paper.

    diff, v9, current

    • CommentRowNumber7.
    • CommentAuthorThomas Holder
    • CommentTimeOct 7th 2019

    Added a reference to .

    diff, v10, current

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 25th 2020

    Added two references

    diff, v13, current

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 30th 2020

    Added this example

    diff, v14, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 1st 2020

    Added two examples: Stone spaces and sober spaces.

    diff, v16, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2021

    added to the list of examples a brief pointer to the inclusion of finite-homotopy-groups homotopy types, as per the new section there

    diff, v20, current

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 23rd 2021

    Added some more examples.

    diff, v23, current

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 26th 2022


    • Ruben Van Belle, Probability monads as codensity monads. Theory and Applications of Categories 38 (2022), 811–842, (tac)

    diff, v25, current

    • CommentRowNumber14.
    • CommentAuthorvarkor
    • CommentTimeFeb 26th 2023

    Point out that every monad (in V-Cat) is a codensity monad.

    diff, v26, current

    • CommentRowNumber15.
    • CommentAuthorvarkor
    • CommentTimeMar 9th 2023

    Mention codensity monad of Yoneda embedding.

    diff, v27, current

    • CommentRowNumber16.
    • CommentAuthorThomas Holder
    • CommentTimeMar 10th 2023

    added a refernce to

    • Anders Kock, Continuous Yoneda Representations of a Small Category, Preprint Aarhus University (1966).(pdf)

    diff, v28, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2023

    added publication data for this item:

    diff, v29, current

    • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023

    Added the observation that the notion makes sense generally in bicategories, and added to the example list what this looks like in RelRel, mentioning this business about specialization topology. While I was at it, I reformatted the list of bullet points into example environments.

    diff, v30, current

    • CommentRowNumber19.
    • CommentAuthorvarkor
    • CommentTimeJun 20th 2023
    • (edited Jun 20th 2023)

    Mention double dualisation as an example.

    Despite this being an “obvious” example, I struggled to find an explicit reference to this fact in the literature. If anyone knows one, please do add a reference.

    diff, v31, current

    • CommentRowNumber20.
    • CommentAuthorThomas Holder
    • CommentTimeAug 9th 2023

    Having recently stumbled on what-is-the-point-of-pointwise-kan-extensions

    I now have some nagging doubts whether the mere existence of Ran here suffices to yield a monad structure for general non pointwise Ran. Actually, the only quick reference I can find for this is exercise 3.a) p.250 of MacLane’s textbook.

    Hence for the moment I put the condition of pointwise Kan extension in the definition, concordant with most of the literature and the fact that Kan extensions in the wild are pointwise anyway.

    diff, v32, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2023

    Have further adjusted the wording in the very first two sentences (here), for clarity and flow.

    diff, v33, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2023

    also streamlined wording, typesetting and hyperlinking throughout the Definition-section (here)

    diff, v33, current

    • CommentRowNumber23.
    • CommentAuthorvarkor
    • CommentTimeAug 9th 2023

    Re. #20. Pointwiseness is not necessary (see section 2 of Street’s “The formal theory of monads”, for instance), but it is present in almost all examples in practice.

    • CommentRowNumber24.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 9th 2023

    Re #20: I had begun writing out a proof, but I’m still sorting out my tikzcd code. Maybe the problem is that I don’t know how to interpret “The only difference to LaTeX is that \usetikzlibrary lines should be put inside the blocks” in HowTo.

    • CommentRowNumber25.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 9th 2023

    I guess better now, although I still don’t understand that HowTo instruction. Taking a break for now.

    diff, v35, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2023

    I don’t know about loading tikzlibraties for the tikz-rendering on nLab pages.

    (If I need them for a diagram then I render that diagram locally and include it on the nLab as an image.)

    Maybe somebody else here knows. Otherwise you’ll have to ask Richard Williamson, who implemented the tikzfunctionality here.

    • CommentRowNumber27.
    • CommentAuthorThomas Holder
    • CommentTimeAug 9th 2023
    • (edited Aug 9th 2023)

    Maybe I misread the [MO example]( what-is-the-point-of-pointwise-kan-extensions) then, but the dual of the non dense functor with Lan the identity functor there yields a non codense functor with a trivial codensity monad, doesn’t it!? Showing at least that the terminology is akward for non pointwise Ran and clashing with the claim that triviality of the codensity monad is equivalent to codensity of the generating functor.

    But thanks for clarifying the issue, anyway!

    • CommentRowNumber28.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 9th 2023

    Re #26: it seems all I needed to do is remove all the \[...\]\backslash [ ... \backslash ] that I had in my code, and this should be reflected in the HowTo. This (to me) mysterious “The only difference to LaTeX is that \usetikzlibrary lines should be put inside the blocks”, where the “should” reads as an instruction, appears to be unneeded and unhelpful for the ordinary nLab editor who just wants to produce readable tikz output, and I think the HowTo page ought to be edited there, but I’m not entirely confident about that position.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2023

    I am not sure what you have in mind, but feel invited to edit the HowTo page.

    • CommentRowNumber30.
    • CommentAuthorvarkor
    • CommentTimeAug 9th 2023

    Re. #27. While it’s true that the Kan extension of a functor along itself always defines a monad, it’s true that this may be trivial without the functor being codense, if the Kan extension is nonpointwise. Therefore, it might be better only to use the terminology “codensity monad” for the pointwise case (indeed, some authors appear to do this), but mention that the construction works without pointwiseness.

    • CommentRowNumber31.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 9th 2023

    I added a few words in response to varkor’s last comment.

    diff, v38, current

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2023

    Best to reference these claims.

    • CommentRowNumber33.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 9th 2023

    Put in the remaining details of the proof that Ran GGRan_G G forms a monad.

    diff, v39, current

    • CommentRowNumber34.
    • CommentAuthorRodMcGuire
    • CommentTimeAug 9th 2023

    added publication data

    diff, v40, current

    • CommentRowNumber35.
    • CommentAuthorThomas Holder
    • CommentTimeAug 11th 2023

    Added a reference to

    as well as

    that I copied over from Myles Tierney. Is there any reason that the link goes to the very likely pay-walled publisher pdf rather than the free version in the TAC-reprint which is presumably checked for typos and looks much nicer?

    Well, anyway: Now that Todd has restored my faith in the general definition of the monad I must confess my nagging doubts are replaced by a strong itch to revert back to the general definition - nevermind the misleading terminology. For the moment I can fight off this urge since the current definition has the charm that it conforms to claims here and at codense functor about the connection to codensity.

    diff, v42, current

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeAug 11th 2023

    Is there any reason

    Such ontological questions are hard to answer satisfactorily.

    But I have now hyperlinked the article to Seminar on Triples and Categorical Homology Theory.

    Also added pointer to:

    diff, v43, current

    • CommentRowNumber37.
    • CommentAuthorThomas Holder
    • CommentTimeAug 26th 2023

    Added a refernce to

    • Ralf Hinze, Kan extensions for program optimisation - Or: Art and Dan explain an old trick, in: Jeremy Gibbons, Pablo Nogueira (eds.), 11th International Conference on Mathematics of Program Construction (MPC ’12), LNCS 7342 Springer (2012) 324–362. (doi: 10.1007/978-3-642-31113-0_16, pdf draft)

    and expanded a bit on the relational example.

    diff, v45, current

    • CommentRowNumber38.
    • CommentAuthorvarkor
    • CommentTimeMay 12th 2024

    Clarify the double dualisation example.

    diff, v46, current