On property-like structures

Steve Lack – 9 April 1997

This was a report on joint work with Max Kelly; a preprint will very soon be available. The purpose of the talk was to describe certain differences between two 2-monads on Cat: the one whose algebras are monoidal categories, and the one whose algebras are categories with finite coproducts. Whereas for the first, a given category can have many algebra structures; for the second such structure is ``essentially unique'' if it exists. Whereas for the first, a functor between monoidal categories can have many structures of monoidal functor; for the second, a functor between categories with finite coproducts either preserves finite coproducts (in the usual sense) or does not. Thirdly any functor between categories with finite coproducts has exactly one structure of lax morphism of categories with finite coprodcuts; this is not at all the case for functors between monoidal categories. We propose a general framework in which to state conditions involving existence and/or uniqueness of T-algebra or T-morphism structure for a 2-monad T on a 2-category K. We call such a 2-monad property-like if algebra structure is essentially unique and morphism structure is unique; this is in fact equivalent to essential uniqueness of ``generalized algebra structure'', by which we mean an action of T not an object of K but on a 2-functor with codomain K. We call such a 2-monad lax-idempotent if every arrow between T-algebras admits a unique structure of lax T-morphism; the lax-idempotent 2-monads turn out to be precisely those 2-monads (T,m,i) for which m is left adjoint to iT with the counit being an identity. These are the strict version of the doctrines studied by Kock, Zoberlein, and others, and provide a very good ``lax'' version of the notion of idempotent monad. (Note added 15/01/98) This work has now been published as On property-like structures, Theory and Applications of Categories 3(1997), 213-250.