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.