On flexibility for 2-monads
by
G. M. (Max) Kelly
University of Sydney
Coauthors: Steven Lack (University of Sydney), A. J. (John) Power

For a 2-monad T (in the strict sense) on a 2-category K, we consider the relation between (strict) T-algebras and pseudo-T-algebras. In the cases of interest it turns out that pseudo-T-algebras are just the T'-algebras for a new 2-monad T' on K. If q: T' --> T is the (strict) morphism of 2-monads inducing the inclusion of the T-algebras into the T'-algebras, the 2-monad T is said to be flexible when qr = 1 for some (strict) morphism r: T --> T' of 2-monads; in which case rq is isomorphic to 1, so that q and r are mutually-inverse equivalences. We show that, in the 2-category of 2-monads on K and (strict) morphisms of these, the flexible 2-monads are closed under flexible colimits; whence we conclude that, when K is the 2-category of small categories or small groupoids (for instance), T is flexible if the structure of a T-algebra can be so presented, in terms of basic operations and equations, that there are equations only between arrows, and not between objects. We go on to observe that, when T is flexible, T-algebra structure can be transported along an equivalence; and we end with an application of this to coherence results. (A small part of these results was presented at the 1995 Cambridge meeting, but was never published.)

Date received: June 14, 2002