Re: Combining monads

[categories <cat-dist@mta.ca>]

Date: Thu, 15 Jan 1998 23:21:27 +0100
From: Jan Juerjens <juerjens@Informatik.Uni-Bremen.DE>
 
> 
> 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 ... :-) ]