added end formula

]]>Clarify the double dualisation example.

]]>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.

]]>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:

- Fred Linton,
*Codensity triples*, Section 8 in:*An outline of functorial semantics*, in*Seminar on Triples and Categorical Homology Theory*, Lecture Notes in Mathematics**80**, Springer (1969) 7-52 [doi:10.1007/BFb0083080]

Added a reference to

- MO-discussion Tim Campion: What is the point of pointwise Kan extensions?

as well as

- Harry Applegate, Myles Tierney,
*Categories with models*, p. 156-244 In: Beno Eckmann (ed.)*Seminar on Triples and Categorical Homology Theory*Lecture Notes in Mathematics,**80**, Springer 1969 (doi:10.1007/BFb0083086, pdf)

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.

]]>added publication data

- {#Sipos} Andrei Sipoş,
*Codensity and Stone spaces*, Mathematica Slovaca, 68 no. 1, p. 57–70, (2018). doi:10.1515/ms-2017-0080, (arXiv:1409.1370)

Put in the remaining details of the proof that $Ran_G G$ forms a monad.

]]>Best to reference these claims.

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

]]>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.

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

]]>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.

]]>Maybe I misread the [MO example](https://mathoverflow.net/questions/220246/ 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!

]]>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.

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

]]>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.

]]>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.

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

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

]]>Having recently stumbled on

https://mathoverflow.net/questions/220246/ 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.

]]>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.

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

]]>added publication data for this item:

- Tom Avery,
*Codensity and the Giry monad*, Journal of Pure and Applied Algebra**220**3 (2016) 1229-1251 [arXiv:1410.4432, doi:10.1016/j.jpaa.2015.08.017]

added a refernce to

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