Fred, Thanks. My 'intuition' is that there are some adjoint decompositions that genuinely reveal internal structure of the monad and others that ... make no such disclosure ;-). But, perhaps i can disabuse myself of this with some application to calculation. Best wishes, --greg On 9/13/07, Fred E.J. Linton <fejlinton@usa.net> wrote:
Greg Meredith asks,
... are there monads such that the only adjoint situations giving rise to them are the Kleisli and Eilenberg-Moore algebras?
NOt even the identity monad on SETS has this property, as it is the adjunction monad also for the adjoint pair
[underlying pointset]: [topological spaces] --> SETS , [discrete topology on]: SETS --> [topological spaces] .
There ARE a few monads for which the Kleisli and E-M categories "coincide," however, beyond the identity monads. First example coming to mind is the FreeVectorSpace monad on SETS. I'm sure other Categories-readers will point out more.
-- Fred