categories: Re: Kleisli and colimits

[Carsten Fuhrmann <C.Fuhrmann@cs.bham.ac.uk>]

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