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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 15th 2010

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 15th 2010

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

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeNov 17th 2010

Yes, that seems right to me.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeNov 17th 2010

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 ?

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeNov 17th 2010

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.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeNov 17th 2010

Thanks for more detailed/precise statement, Todd.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeDec 6th 2010

1. 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

• CommentRowNumber9.
• CommentAuthorJohn Baez
• CommentTimeFeb 15th 2021

• CommentRowNumber10.
• CommentAuthorvarkor
• CommentTimeSep 16th 2021

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

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeSep 16th 2021

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