Maps of monads - references
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
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
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
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
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
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
participants (5)
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Michael Barr -
Prof. Dr. Pumpluen -
Steve Vickers -
Steven R. Costenoble -
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