Re: Combining monads

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

Date: Thu, 15 Jan 1998 15:44:30 +1100 (EST)
From: Steve Lack <stevel@maths.usyd.edu.au>

> 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.