On property-like structures

G. M. Kelly and Stephen Lack

A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of `category with finite products'. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of `essentially unique' and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which `structure is adjoint to unit', and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads.