Rereading my message, I realized I should perhaps clarify that not just the name, but (as far as I know) the concept itself was originated by Kelly and Lack. The example of semigroups is also in their paper, which is concerned mainly with the case when the forgetful functor is also 2-monadic. The resulting "property-like 2-monads" generalize both "lax-idempotent" (= "Kock-Zoberlein") 2-monads, such as those which assign colimits, and the dual "colax-idempotent" 2-monads, such as those which assign limits. But they are strictly more general than either: for instance, a 2-monad which assigns both limits and colimits is property-like, but not lax- or colax-idempotent. Mike On Sun, Oct 3, 2010 at 6:32 AM, Colin McLarty <colin.mclarty@case.edu> wrote:

I like this discussion by Mike Shulman.  And a propos of the related discussion of terminology I note the terms here describe mathematical features (very well, I think) rather than focusing on whether one *likes* the features.

2010/10/2 Michael Shulman <shulman@math.uchicago.edu>:

...

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.  In terms of forgetful functors, property-like structure corresponds to a functor which is *pseudomonic*, i.e. faithful, and full-on-isomorphisms.  Another nice example is that being a monoid is a "property" of a semigroup, i.e. a semigroup can have at most one identity element, but a semigroup homomorphism between monoids need not be a monoid homomorphism.

best, Colin

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