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The link for ’equivalent’ at the top redirected to natural isomorphism which (as I understand it) is the correct 1-categorical version of an equivalence of functors, but this initially lead me to believe that a functor was monadic iff it was naturally isomorphic to a forgetful functor from the Eilenberg-Moore category of a monad on its codomain, which would mean that the domain of the functor was literally the Eilenberg-Moore category of some adjunction since natural isomorphism is only defined for parallel functors.
I also added a few words in the definition section that seemed to be missing, apologies for the double post.
No problem, I have tidied up the postings :-).
Much appreciated!
added some references:
Francis Borceux, Def. 4.4.1 in: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]
Emily Riehl, §5.3 in: Category Theory in Context, Dover Publications (2017) [pdf]
I have hyperlinked all occurrences of “comparison functor” on this page, and am giving this is own page now.
added the proposition (here) that monadic functors create limits
(copied over from the created limit)
The link to 1lab here and at reflective subcategory is broken; I don’t know whether it was intended to link directly to the relevant page or to a reference at the bottom of the page.
Thanks for the heads up. After a little inspection, I see that something strange has been happening:
The reference which the seemingly broken link points to is certainly present in the source code: see the very bottom of ncatlab.org/nlab/source/reflective+subcategory.
But with revision 111 by “Gnampfissimo” the whole last third of the entry (most of the Examples-section, and all of the References) is marked in the logs as deleted and indeed no longer shows in the rendered page.
Nevertheless, the code for all the seemingly deleted material is still visible in the source.
So what gives?
Turns out that this revision 111 by “Gnampfissimo” also changed
+-- {: .num_example }
###### Example
A category of [[concrete presheaves]] inside a [[category of presheaves]] on a [[concrete site]] is a reflective subcategory.
=--
to
+-- {: .num_example }
###### Example
A category of [[concrete presheaves]] inside a [[category of presheaves]] on a [[concrete site]] is a reflective subcategory.
=--apocalypticists
and apparently this triggers a bug in the rendering software which causes all following code to be ignored.
I am not fixing this right now. Will first bring this in situ to the attention of technical team.
Have fixed now the entry reflective subcategory, see the discussion there.
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