Date: Wed, 14 Jan 1998 16:21:51 +0000 (GMT) From: Tom Leinster <T.Leinster@dpmms.cam.ac.uk>
Is the pullback of a monadic functor along a monadic functor necessarily monadic? Is the diagonal of the pullback square monadic? Does this work if your restrict yourself to, say, finitary monadic functors?
(E.g. it works for finitary monads on Set: the theory of sets with both ring and lattice structure (not interacting in any particular way) comes from a monad.)
Thanks, Tom Leinster
Hi Tom, if I'm not mistaken, this reduces for full isomorphism-closed embeddings to the (finite) Intersection Problem (of full iso-closed subcategories) answered negatively by Trnkova, Adamek, Rosicky ("Topological reflections revisited", ProcAMS 108,3 (1990) p605; see also Tholen "Reflective Subcategories" TopAppl 27 (1987) p201, Adamek, Rosicky "Intersections of reflective subcategories" ProcAMS 103 (1988) p710). Full iso-closed subcategories of locally lambda-presentable categories are reflective and closed under lambda-directed colimits iff they are lambda-orthogonal, so intersections of such subcategories are reflective (Adamek, Rosicky "Locally presentable and Accessible Categories" CUP 94). Bye, Jan [ + thanks again for the supervisions ... :-) ]