On Property-like Structures
نویسنده
چکیده
A category may bear manymonoidal structures, but (to within a unique isomorphism) only one structure of \category with nite 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 de nition 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 laxmorphisms 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 nitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each core ective among all 2-monads.
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