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Does the dot $F \cdot G$ between two functors indicate composition in traditional Leibnizian order,or does it mean “$F$ followed by $G$”? Because if the former, I think the universal arrow ought to be instead $\alpha: (Ran_G G) \cdot G \Rightarrow G$.
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.
Added two references
Jirí Adámek, Lurdes Sousa, D-Ultrafilters and their Monads, (arXiv:1909.04950)
Andrei Sipos, Codensity and Stone spaces, (arXiv:1409.1370)
Added this example
added a refernce to
added publication data for this item:
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.
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.
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.
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.
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.
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!
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.
I am not sure what you have in mind, but feel invited to edit the HowTo page.
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.
Best to reference these claims.
added publication data
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.
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:
Added a refernce to
and expanded a bit on the relational example.
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