Defining a morphism <U,f> from an monad <S,e,m> on X to a monad <S',e',m'> on X' in such a way that U is a 1-cell (functor) from X to X', there are two choices for the direction of the 2-cell (nat. trans.) f. Street in "The formal theory of monads", JPAA 2(1972), 149--168, chooses f to go from the composit of U with X' to the composit of X with U, while Barr and Wells in TTT choose the other direction (although in their case X=X'). The second choice seems more intuitive to me, since the appropriate specialization gives ordinary monoid homomorphisms. What is the rationale for the first choice? J"urgen Koslowski Dept. of Math. Dept. of Comp. & Info. Sci. Kansas State University =========================================================================
I believe Street defined both kinds of morphism, calling them "monad functors" and "monad opfunctors". The Eilenberg-Moore (category of algebras) construction is functorial with respect to the functors (this is touched on in MacLane's Categories for the Working Mathematician, exercise VI.2.3). The Kleisli construction is functorial with respect to the opfunctors. Steve Vickers. =========================================================================
Defining morphisms between monads in a 2-category C (where C = Cat if you like) the way I do in "The formal thy of mnds", I obtain a 2-category Mnd(C) of monads in C such that the inclusion C --> Mnd(C) has the Eilenberg-Moore construction as its right adjoint. On page 159 of the paper, J"urgen will find the 2-cell which satisfies his intuition; these are good for Kleisli's construction. Of course, we now know that the EM-construction is just a weighted (or "indexed") limit in the Cat-enriched context ("Limits indexed by cat-valued 2-functors" JPAA 1976?). Kleisli-construction is a weighted colimit. Also see SLNM 420 (Kelly-Street paper and "Elem cosmoi" Section 6 pp166-168). Regards, Ross =========================================================================
participants (3)
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koslowj＠cis.ksu.edu -
Steven John Vickers -
street＠macadam.mpce.mq.edu.au