One construction on lax functors
Steve Lack – 13 November 1996
If T is a 2-monad on a 2-category K then one can define the 2-category T-Alg of (strict) algebras, pseudomorphisms, and algebra 2-cells for the 2-monad. It has a locally full sub-2-category T-Alg_s with the same objects but with only the strict morphisms as arrows. Provided, say, that K is locally presentable and T has a rank, the inclusion J of T-Alg_s in T-Alg is known to have a left adjoint, sometimes called ( )'. In this talk I give a construction of the left adjoint in terms of iso-coinserters and coequifiers in T-Alg_s; once again, these colimits in T-Alg_s are known to exist if K is locally presentable and T has a rank.
A natural context for the above construction is that of a sufficiently cocomplete 2-category L (which takes the place of T-Alg_s) with a 2-comonad on it (taking the place of the comonad induced by the adjunction between T-Alg_s and K). In this generality one can produce a new comonad on L; forming the Kleisli object for this comonad recovers, in the case of the previous paragraph, the 2-category T-Alg.
Dually one could start with a 2-monad on a sufficiently complete 2-category and produce a new 2-monad, and then consider its Kleisli object. In fact a 2-monad T is just a lax functor (of 3-categories) from 1 to 2-Cat, and this construction can be carried out on more general lax functors, whence the title of the talk. For those tickled by the idea of a second construction on lax functors, one can repeat the whole analysis considering the lax rather than the pseudo morphisms of algebras.
Returning to the construction of ( )' in the first paragraph, one sees that every T-algebra which is flexible in the sense of Bird, Kelly, Power, Street, Flexible limits for 2-categories, JPAA 61 (1989), 1-27 is a flexible colimit in T-Alg_s of free algebras. In that same paper, the authors prove that a flexible colimit (in T-Alg_s) of free algebras is flexible, and so we have the result that the flexible algebras for a 2-monad are precisely the closure under flexible colimits in T-Alg_s of the free algebras.
Finally I observed that this old result that flexible colimits of flexible algebras are flexible gives an elegant and general proof of the theorem of Crans-Kelly-Power (paper in preparation) that ``monads given by inductive presentations with no equations between objects are flexible''. In the C-K-P paper, various operations on monads are seen to preserve flexibility, and these operations can all be described as forming certain flexible colimits in the 2-category 2-Mnd(K) of 2-monads on K and strict maps of such.
At the end of the talk, Michael Batanin pointed out some connections between the construction of ( )' and his paper Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers de Topologie et Géometrie Différentielle, XXXIV-4(1993), 279-304.