Date: Wed, 14 Jan 1998 15:12:36 -0400 (AST) Subject: Combining monads Date: Wed, 14 Jan 1998 16:21:51 +0000 (GMT) From: Tom Leinster 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 19:40:09 -0400 (AST) Subject: Re: Combining monads Date: Thu, 15 Jan 1998 09:58:50 +1100 From: Ross Street >Date: Wed, 14 Jan 1998 16:21:51 +0000 (GMT) >From: Tom Leinster > > >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 19:47:30 -0400 (AST) Subject: oops Date: Thu, 15 Jan 1998 10:32:43 +1100 From: Ross Street I just realised my hasty response to Tom Leinster was to the question of pullback of a monadic alond an *arbitrary* functor; not Tom's question! --Ross Date: Thu, 15 Jan 1998 17:10:42 -0400 (AST) Subject: Re: Combining monads Date: Thu, 15 Jan 1998 15:44:30 +1100 (EST) From: Steve Lack > Date: Wed, 14 Jan 1998 15:10:43 -0400 (AST) > From: categories > > Date: Wed, 14 Jan 1998 16:21:51 +0000 (GMT) > From: Tom Leinster > > > 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: Fri, 16 Jan 1998 14:21:02 -0400 (AST) Subject: Re: Combining monads Date: Thu, 15 Jan 1998 23:21:27 +0100 From: Jan Juerjens > From cat-dist@mta.ca Wed Jan 14 20:08:51 1998 > Date: Wed, 14 Jan 1998 15:10:43 -0400 (AST) > From: categories > To: categories > Subject: Combining monads > MIME-Version: 1.0 > > Date: Wed, 14 Jan 1998 16:21:51 +0000 (GMT) > From: Tom Leinster > > > 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 ... :-) ] Date: Tue, 20 Jan 1998 15:03:21 -0400 (AST) Subject: Re: pullback categories Date: Tue, 20 Jan 1998 14:20:35 -0500 (EST) From: Michael Barr Triples is back, at least for the time being, after having been down, either from its problems or the lack of power at McGill for most of the last month. I have just seen the question about pullback of tripleable functors and I have not seen a really satisfactory reply. A lot--maybe all--of what I say below is probably in Ernie Manes' thesis (A Triple Miscellany, Wesleyan U., 1967). The answer is definitely no, but the problem is the lack of adjoint. Consider, for the example, the complete semilattice triple on Set. It is also the covariant powerset triple, with singleton for eta and union (or intersection) for mu (one will give you sup semilattices, the other the inf semilattices). That is one triple and the other is N x -, whose algebras are sets with a single unary operation. The pullback is the category whose objects are complete semilattices with a single unary function that is not assumed to cohere in any way with the semilattice structure. A complete boolean algebra is model of such a theory, taking complement as the unary operation. If there were free algebras of this type, then a quotient of them would be a free complete boolean algebra, but we know these don't exist. Of course, one can raise the question under the assumption that the adjoint exists. For example, if the category is locally presentable (complete and accessible) and the functors are accessible. Then if the functors are T_1 and T_2, simply apply them alternately, taking colimits at limit ordinals, until they stabilize and that will give you free algebras. Of course, it is obvious that the pullback is the category whose objects consist of T_1A ---> A <--- T_2A which are algebras for each triple, but no assumption of coherence between the two structures is made. It seems pretty obvious, although I have no really checked the details carefully, that this will satisfy Beck's condition. Or rather the forgetful functor to each of the individual category of algebras as well as the composite. For example, if we had the situation T_1A -----------> A <------------- T_2A || || || || || || || || || || || || || || || || || || vv vv vv T_1B -----------> B <------------- T_2B of such a nature that A ======> B -----> C is a split coequalizer, then the outer columns of T_1A -----------> A <------------- T_2A || || || || || || || || || || || || || || || || || || vv vv vv T_1B -----------> B <------------- T_2B | | | | | | | | | | | | | | | | | | v v v T_1C - - - - - -> C <- - - - - - - T_2C are split coequalizers too, whence the dotted arrows exist. That the whole diagram is a coequalizer in the pullback category is also easy. Thus the answer is yes, provided the adjoint exists, but that is not guaranteed even over Set. Michael Date: Tue, 20 Jan 1998 17:34:05 -0400 (AST) Subject: Re: Combining monads and Pullback cats Date: Tue, 20 Jan 98 15:30:07 EST From: John Duskin Mike is right. The problem lies in the existence of a left adjoint. However if one pulls back over a Grothendieck co-(=op-)fibration, then if U has a left adjoint, then so does its pullback. In fact, a functor is a cofibration iff it is universal for this property, i.e., it has this property and this property is stable under arbitrary pullbacks."A cofibration is a universal change of base for right adjoints". I don't know if this is of much help, but it is amusing.