Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
Just for definiteness I stated (again) at conservative functor the property that such reflects all (co)limits wich it preserves
Yes, that seems right to me.
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 ?
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.
Thanks for more detailed/precise statement, Todd.
I added a couple more examples to monadicity theorem.
1 to 9 of 9