Date: Wed, 14 Jan 1998 15:10:43 -0400 (AST) From: categories <cat-dist@mta.ca>

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

Let K be a complete and cocomplete category, and Mnd(K) the category of monads on K and strict morphisms of monads. If T and S are monads on K which preserve (alpha-)filtered colimits (for a regular cardinal alpha), then (i)the coproduct T+S exists in Mnd(K) (ii)this coproduct is ``algebraic'', meaning that the diagonal of the pullback square K^S | | v K^T-->K is the forgetful functor K^(T+S)-->K (iii)the projections K^(T+S)-->K^T and K^(T+S)-->K^S are monadic. Much can be done without completeness, but the proofs become a bit harder. See the paper G.M.Kelly, A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, Bull. Austral. Math. Soc. 22(1980):1--83 for a survey of many such results. In fact if K is locally finitely presentable then the category Mnd_f(K) of finitary monads on K and strict morphisms of monads is itself locally finitely presentable; for this see my paper ``On the monadicity of finitary monads'', to appear in JPAA, but in the meantime available at http://www.maths.usyd.edu.au:8000/res/Catecomb/Lack/1997-29.html. Regards, Steve.