Gaunce Lewis wrote:
I have encountered a situation in which I have two categories C, D which are related by a pair of adjoint functors L from C to D and R from D to C. Also, there is a cotriple S on C and a cotriple T on D. Finally, there is a natural isomorphism f from RT to SR. It seems that if a couple of diagrams relating f to the structure maps of the cotriples commute, then there is an induced adjoint pair relating the two coalgebra categories. Is this, or something similar to it, in the literature in some easily referenced place?
Thanks, Gaunce
Here's a related reference: the appendix of C.Hermida and B.Jacobs, Structural Induction and Coinduction in a fibrational setting, Information and Computation 145(2) 107-152,1998. stablishes the suitable 2-functoriality of categories of (co)algebras for endofunctors as inserters (which does not follow straightforwardly from their weighted limit formulation) and the more or less immediate corollaries of induced adjoints, without any coequalisers or additional structure. It is remarkably simple but fairly useful. The result (and the argument) extends literally to the case of Eilenberg-Moore algebras for monads: using the (old) notion of morphism of monads from (a), given a pseudo-morphism of monads (f,\theta):M -> N (where \theta is iso), if f has a right adjoint g, adjoint transposition of \theta yields (g,\theta'):N -> M right adjoint to (f,\theta) in Mnd (b), and 2-functoriality of algebras yields the desired adjunction between the categories of algebras (commuting with the forgetful functors, of course). Once again, no structure is required on the categories involved. (a) R. Street, The formal theory of monads, JPAA 2 (1972) 149-168 (b) R. Street, Two constructions on lax functors, Cahiers top. et geom. diff. 13 (1972) 217-264. Claudio PS: There is a more liberal notion of 2-cell for Mnd, essentially arising from internal category theory, but I don't know its impact in the above adjoint results.