Dear Fred, If we are talking about funny things - all right, let us talk about funny things: I claim that there is exactly one pair (C,T) in which: 1. C is a category, and T is a monad on C; 2. the only adjoint situations giving rise to T are determined by the Kleisli and the Eilenberg-Moore categories. Even though "unfortunately", in this case the Kleisli and the Eilenberg-Moore categories coincide. In this unique pair C is the empty category. Indeed, if C is non-empty, then take any category A with zero object and look at the projection AxAlg(T) ---> Alg(T) composed with the forgetful functor Alg(T) ---> C. This composite together with its obvious left adjoint will give rise to T. Just coincidence of the Kleisli and the Eilenberg-Moore categories is another story of course... George Janelidze ----- Original Message ----- From: "Fred E.J. Linton" <fejlinton@usa.net> To: <categories@mta.ca> Sent: Friday, September 14, 2007 12:50 AM Subject: categories: Re: "prime" monads? 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