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

kevindcarlson

]]>added DOI-s and other missing publication data,

and copied these items also to their respective author pages

]]>Added two more relevant references of Linton.

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

]]>Added some references.

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

]]>Corrected section number in the reference to Lurie HA.

Mark Barbone

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

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

]]>added also mentioning of the case of monoid actions: here

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

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

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

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

]]>Added a bit more info on crude monadicity.

]]>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

]]>I added a couple more examples to monadicity theorem.

]]>Thanks for more detailed/precise statement, Todd.

]]>In other words, the proposition says that if $lim K$ exists in $C$, and if $U: C \to D$ is a conservative functor such that $U(lim K)$ is the limit of $U K$ in $D$, and if $const_c \to K$ is a cone for which the induced cone $U c \to U K$ is the limit, then $c$ is the limit of $K$. So one of the hypotheses is that the limit exists in the subcategory.

]]>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 ?

]]>Yes, that seems right to me.

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

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

$C$ has

I think it must be

$D$ has

and have changed it accordingly. But have a look.

]]>