TITLE: Completeness and cocompleteness of the Kleisli category
QUESTION [10 upvotes]: If (T,η,μ) is a monad over a category C, which is complete and cocomplete, then what about the Kleisli category? And also, if C is cartesian closed, what about the Kleisli one?

REPLY [5 votes]: The article on limits and colimits in the Kleisli category shows that (with $C_T$ denoting the Kleisli category, and $C^T$ denoting the Eilenberg-Moore category):
1) If $C$ is complete, and the canonical functor $C_T\to C^T$ has a right adjoint, then $C_T$ is complete. 2) If $C$ has coproducts, then $C_T$ has coproducts (already mentioned in the comments to another answer to the question). 3) Cocompleteness is more subtle, and under some condition on partial algebras cocompleteness of $C_T$ is equivalent to the existence of a left adjoint of the canonical functor $C_T\to C^T$.