Does anyone know if the Kleisli construction (sending a monad to its Kleisli category) behaves in any decent way with respect to colimits? E.g. does it in any sense preserve or reflect them? The actual situation that I have is a fixed category C, and a certain coequalizer diagram in the category of monads on C. The resulting fork in Cat is also a coequalizer, and the proofs that both diagrams are coequalizers have some ingredients in common, but I can't at present see how to deduce one from the other. Thanks, Tom Leinster
The construction of the Kleisli category can be turned into a left adjoint functor whose domain is the category of Mnd monads (on arbitrary categories) and monad morphisms. However, the codomain of this functor is not Cat, but a category AbsKl whose objects I call "abstract Kleisli categories". An abstract Kleisli category is a category K together with a functor L:K->K, a natural transformation \epsilon: L->Id, and a (not generally natural) transformation \theta:Id->L, such that (1) \theta_L is a natural transformation (2) L\theta o \theta = \theta_L o \theta (3) \epsilon o \theta = id (3) L\epsilon o \theta_L = id A morphism K->K' of AbsKl is a functor that preserves the solutions of the non-naturality square \theta o f = Lf o \theta (I) The Kleisli construction forms a functor Mnd->AbsKl. Its right adjoint sends an abstract Kleisli category K to the evident monad on the subcategory given by the solutions of Equation (I). The counit of this adjunction is in fact an iso, so AbsKl is a full reflective subcategory of Mnd. The full subcategory of Mnd which is equivalent to Abskl is given by those monads for which every component of the unit is a regular mono. Cheers, Carsten
participants (2)
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Carsten Fuhrmann -
Tom Leinster