Let $C$ be a cocomplete category, and suppose that it has an object that is colimit dense. Is $C$ automatically monadic over $Set$? And if not, is there an explicit counterexample?



Colimit density and monads
I guess I'll go out on a limb and assume arsmath means, when he says an object $c$ of $C$ is colimit-dense, that the full subcategory containing $c$ is a dense subcategory in the [usual sense](http://ncatlab.org/nlab/show/dense+functor). 


There's some categorical lore which is helpful: a necessary condition for a category $C$ to be monadic over $Set$ (that is, for there to exist some monadic functor $U: C \to Set$) is that $C$ be a [regular category](http://ncatlab.org/nlab/show/regular+category). I think this result is proved somewhere in Francis Borceux's Handbook of Categorical Algebra, but the quickest reference I can find to hand is [here](http://perso.uclouvain.be/enrico.vitale/monadiche_su_Set.pdf), Proposition 1, part (2) (here $Set$ is a regular category, and any monad $T: Set \to Set$ preserves epis because every epi in $Set$ is already split, by the axiom of choice). 


Now I claim the category $Pos$ of posets is not regular, hence cannot be monadic over $Set$. It is not regular because the pullback of a regular epi in $Pos$ need not be a regular epi; see the explicit example described [here](http://ncatlab.org/nlab/show/regular+category#examples). On the other hand, it isn't too hard to see that $Pos$ is cocomplete and the two-element chain $\{0 \leq 1\}$ is colimit dense in $Pos$. So this gives an explicit counterexample. 


