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
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?
No. And I seem to remember this was one of the main points of the thesis (under Lawvere) of Michel Thie'baud. The thesis title was "Self-dual structure-semantics & algebraic categories" (Dalhousie University, Halifax, Nova Scotia, August 1971). Comonads (= cotriples) in Mod (= Bimod = Prof = Dist) are the subject. Using these to define "algebraic", Michel obtained stability under pullback. --Ross
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.
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 ... :-) ]
participants (4)
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Jan Juerjens -
Ross Street -
Steve Lack -
Tom Leinster