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On property-like structures

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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.

Keywords: 2-category, monad, structure, property.

1991 MSC: 18C10,18C15,18D05.

*Theory and Applications of Categories*, Vol. 3, 1997, No. 9, pp 213-250.

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