I had been looking at the standard way the topological closure A |-> CA can be recovered from the derived set operation A |-> A' (which takes A to the set of the accumulation points of A) by setting CA := A' u A, and observed that it can be generalized to categorical language. If you take thing made up of a coproduct-preserving functor D and a natural transformation t : D^2 -> D, such that t is `associative' in the usual sense of asserting that t o t_D = t o Dt, and also p_{X,Y} o t_{X + Y} = (t_X + t_Y) o p_{DX,DY} o Dp_{X,Y} where p is the canonical natural transformation D(X + Y) -> DX + DY , then if one defines TX := DX + X eta_X := inr_{DX + X} mu_X := ([1_{DX},1_{DX}] + X) a_{DX,DX,X} (([t_X,1_{DX}] o p_{DX,X}) + TX) (where [ , ] is copairing and a_{X,Y,Z} is the associativity isomorphism X + (Y + Z) -> (X + Y) + Z) then turns out that (T, eta, mu) is a monad. Has anyone already studied these? I'd be particularly interested in a type-theoretic or logical (along the lines of the correspondence between monads and modal operators) interpretation of such a structure, if any. ---Jason