Re: Maps of monads - references

Re: Maps of monads - references

[Steven R. Costenoble]

Thanks, all, for the many replies (private as well as to the list).
Among other interesting references, almost everyone suggested Ross
Street's "The formal theory of monads" as well as the recent followup
by Street and Steve Lack, "The formal theory of monads II." I'll add
those to my summer reading list.

--Steve Costenoble


On Jul 10, 2007, at 9:25 AM, Steven R. Costenoble wrote:

> In Toposes, Triples, and Theories, Barr and Wells define a morphism
> of triples (which, being a student of Peter May, I will call a map of
> monads) in the context of two monads on a given category C. I have a
> situation where I have two categories C and D, a monad S on C, a
> monad T on D, and a functor F: C -> D. There is a fairly obvious
> generalization of the TTT definition, to say that a map from S to T
> is a natural transformation FS -> TF making certain diagrams commute.
> My guess is that someone else noticed this long ago, so I'm looking
> for references to where this has appeared in the literature. I'm
> particularly interested in references that include the fact (at
> least, I'm pretty sure it's a fact) that such maps are in one-to-one
> correspondence with extensions of F to a functor between the
> respective Kleisli categories of S and T.
>
> Thanks in advance.
>
> --Steve Costenoble
>
>
>

Re: Maps of monads - references

[Prof. Dr. Pumpluen]

There is still another, very detailed reference (probably earlier) to this
topic in my paper "Eine Bemerkung ueber Monaden und adjungierte Funktoren",
Math. Ann.185, 329-337 (1970).

                  Best regards  Nico Pumpluen.

On Jul 10, 2007, at 9:25 AM, Steven R. Costenoble wrote:

> In Toposes, Triples, and Theories, Barr and Wells define a morphism
> of triples (which, being a student of Peter May, I will call a map of
> monads) in the context of two monads on a given category C. I have a
> situation where I have two categories C and D, a monad S on C, a
> monad T on D, and a functor F: C -> D. There is a fairly obvious
> generalization of the TTT definition, to say that a map from S to T
> is a natural transformation FS -> TF making certain diagrams commute.
> My guess is that someone else noticed this long ago, so I'm looking
> for references to where this has appeared in the literature. I'm
> particularly interested in references that include the fact (at
> least, I'm pretty sure it's a fact) that such maps are in one-to-one
> correspondence with extensions of F to a functor between the
> respective Kleisli categories of S and T.
>
> Thanks in advance.
>
> --Steve Costenoble
>
>
>

Re: Maps of monads - references

[Valeria.dePaiva]

There's also:
 The formal theory of monads II, R. Street and S. Lack,J. Pure Appl.
Algebra 175 (1-3) (2002) 243-265; MR2003m:18007 (preprint available from
the homepages of the authors).


Street's work is about 2-categories of monads and many people needed the
simply categorical level, because of the Linear Logic connection. So
there are several printed versions of the restricted result, including
one of our group, I believe, in

Relating Categorical Semantics for Intuitionistic Linear Logic,  (M.
Maietti, P. Maneggia, V. de Paiva and E. Ritter) in Applied Categorical
Structures, volume 13(1):1--36, 2005. 

But given our application we prove it for comonads, lifting both to
Eilenberg-Moore coalgebras and to co-Kleisli categories.

Dr Valeria de Paiva
PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
USA


-----Original Message-----
From: Steve Vickers [mailto:s.j.vickers@cs.bham.ac.uk] 
Sent: Tuesday, July 10, 2007 12:39 PM
To: categories@mta.ca
Subject: categories: Re: Maps of monads - references

See Ross Street "The formal theory of monads", JPAA 2 (1972) 149-168 for
the general definition, in the abstract setting of a 2-category instead
of Cat.

Actually, there are two obvious generalizations of the TTT definition
("monad functors" and "monad opfunctors"), for the two possible
directions of F.

Steve Vickers.

On 10 Jul 2007, at 14:25, Steven R. Costenoble wrote:

> In Toposes, Triples, and Theories, Barr and Wells define a morphism of

> triples (which, being a student of Peter May, I will call a map of
> monads) in the context of two monads on a given category C. I have a 
> situation where I have two categories C and D, a monad S on C, a monad

> T on D, and a functor F: C -> D. There is a fairly obvious 
> generalization of the TTT definition, to say that a map from S to T is

> a natural transformation FS -> TF making certain diagrams commute.
> My guess is that someone else noticed this long ago, so I'm looking 
> for references to where this has appeared in the literature. I'm 
> particularly interested in references that include the fact (at least,

> I'm pretty sure it's a fact) that such maps are in one-to-one 
> correspondence with extensions of F to a functor between the 
> respective Kleisli categories of S and T.
>
> Thanks in advance.
>
> --Steve Costenoble
>
>
>

Re: Maps of monads - references

[Michael Barr]

I think that somewhere Harry Appelgate did something like that, probably
in his Ph.D. thesis.  Whether he ever published it, I cannot now say.

Michael

On Tue, 10 Jul 2007, Steven R. Costenoble wrote:

> In Toposes, Triples, and Theories, Barr and Wells define a morphism
> of triples (which, being a student of Peter May, I will call a map of
> monads) in the context of two monads on a given category C. I have a
> situation where I have two categories C and D, a monad S on C, a
> monad T on D, and a functor F: C -> D. There is a fairly obvious
> generalization of the TTT definition, to say that a map from S to T
> is a natural transformation FS -> TF making certain diagrams commute.
> My guess is that someone else noticed this long ago, so I'm looking
> for references to where this has appeared in the literature. I'm
> particularly interested in references that include the fact (at
> least, I'm pretty sure it's a fact) that such maps are in one-to-one
> correspondence with extensions of F to a functor between the
> respective Kleisli categories of S and T.
>
> Thanks in advance.
>
> --Steve Costenoble
>
>
>

Re: Maps of monads - references

[Steve Vickers]

See Ross Street "The formal theory of monads", JPAA 2 (1972) 149-168
for the general definition, in the abstract setting of a 2-category
instead of Cat.

Actually, there are two obvious generalizations of the TTT definition
("monad functors" and "monad opfunctors"), for the two possible
directions of F.

Steve Vickers.

On 10 Jul 2007, at 14:25, Steven R. Costenoble wrote:

> In Toposes, Triples, and Theories, Barr and Wells define a morphism
> of triples (which, being a student of Peter May, I will call a map of
> monads) in the context of two monads on a given category C. I have a
> situation where I have two categories C and D, a monad S on C, a
> monad T on D, and a functor F: C -> D. There is a fairly obvious
> generalization of the TTT definition, to say that a map from S to T
> is a natural transformation FS -> TF making certain diagrams commute.
> My guess is that someone else noticed this long ago, so I'm looking
> for references to where this has appeared in the literature. I'm
> particularly interested in references that include the fact (at
> least, I'm pretty sure it's a fact) that such maps are in one-to-one
> correspondence with extensions of F to a functor between the
> respective Kleisli categories of S and T.
>
> Thanks in advance.
>
> --Steve Costenoble
>
>
>

Maps of monads - references

[Steven R. Costenoble]

In Toposes, Triples, and Theories, Barr and Wells define a morphism
of triples (which, being a student of Peter May, I will call a map of
monads) in the context of two monads on a given category C. I have a
situation where I have two categories C and D, a monad S on C, a
monad T on D, and a functor F: C -> D. There is a fairly obvious
generalization of the TTT definition, to say that a map from S to T
is a natural transformation FS -> TF making certain diagrams commute.
My guess is that someone else noticed this long ago, so I'm looking
for references to where this has appeared in the literature. I'm
particularly interested in references that include the fact (at
least, I'm pretty sure it's a fact) that such maps are in one-to-one
correspondence with extensions of F to a functor between the
respective Kleisli categories of S and T.

Thanks in advance.

--Steve Costenoble