By definition (at least according to the usage under discussion), something necessarily preserved by all morphisms is a "property," although it can also be regarded as a particular degenerate case of a structure and, I guess, also a degenerate case of a property-like structure. property = forgetful functor is full and faithful structure = forgetful functor is faithful property-like structure = forgetful functor is pseudomonic http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property Mike On Mon, Oct 4, 2010 at 12:52 AM, Vaughan Pratt <pratt@cs.stanford.edu> wrote:
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
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