On 10/2/2010 3:03 PM, Michael Shulman wrote:
I personally prefer to say that "unique choice structure" is something "in between" property and structure. Kelly and Lack dubbed it "Property-like structure" in their paper with that title. The difference is exactly as you say: property-like structure is unique (up to unique isomorphism) when it exists, but is not necessarily "preserved" by all morphisms.
How should this terminology be applied when the property-like structure is necessarily preserved by all morphisms? A group can be defined as a monoid with the property that all of its elements have inverses. The inverse is preserved by all morphisms. A Boolean algebra can be defined as a bounded distributive lattice with the property that all of its elements have complements. The complement is preserved by all morphisms. Are these merely "property-like structures," or are they actual structures, despite being defined merely as properties? Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]