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On limits and colimits in the Kleisli category. (English) Zbl 0533.18006
The author investigates the following problem: Given a monad (triple) \(R=(R,\alpha,\beta)\) on a category \({\mathcal A}\), denote by S the canonical functor from the Eilenberg-Moore category \({\mathcal A}^ R\) to the Kleisli category \({\mathcal A}_ R\). Find conditions on \({\mathcal A}\), \({\mathcal A}^ R\) and S such that the category \({\mathcal A}_ R\) is cocomplete resp. complete.
Results of the following kind are established: The existence of a left- adjoint to S is equivalent to the cocompleteness of \({\mathcal A}_ R\) under certain hypothesis for the categories \({\mathcal A}\) and \({\mathcal A}^ R\). The existence of a right adjoint to S is equivalent to the completeness of \({\mathcal A}_ R\) under certain hypothesis for the category \({\mathcal A}\) and the forgetful functor \(U_ R: {\mathcal A}_ R\to {\mathcal A}.\)
The last part of the paper deals with the existence of a left adjoint to the base functor R of the monad. The results obtained allow a simple proof of a theorem of Tholen concerning the lifting of adjoint functors.
Reviewer: H.Kleisli

MSC:
18C20 Eilenberg-Moore and Kleisli constructions for monads
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
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References:
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