On Thu, 20 May 2004, 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
This situation has been encountered since at least 1970 by various categorists, including myself. A relevant paper is: D. Pumpl\"un, Eine Bemerkung \"uber Monaden und adjungierte Funktoren, Math. Annalen 185, 329-337 (1970). If Gaunce's two commuting diagrams are the usual ones, then his conjecture is correct. Observe that in this situation, we have not just a pair but a quadruple of dual categories, replacing C and D by their duals, or inverting the direction of arrows, or both. This may just be a "folk theorem", but it should have been published by someone, somewhere, and I would also like to have an easily accessible reference, or references. Oswald Wyler