John Baez <b...
@galaxy.ucr.edu> wrote parenthetically:
>leave it to James Dolan to explain the technical distinction between
>"extra properties", "extra structure", and "extra stuff" - there is
>a nice category-theoretic way of making this precise.
Ooh, let me guess!
Given a functor U: C -> D, interpret U as a forgetful functor.
Then C is D with extra *structure* if U is surjective on the objects
and, given a pair of objects, injective on the morphisms between them;
and C is D with extra *properties* if U is injective on the morphisms
(meaning injective on the objects and on the morphisms between a given pair);
Otherwise, I guess C is just D with extra *stuff*
if, given a pair of objects, U is injective on the morphisms between them.
For example, the forgetful functor Groups -> Sets
shows that groups are sets with extra structure,
while the forgetful functor Abelian Groups -> Groups
shows that Abelian groups are groups with extra properties.
Or you can turn around and use the free functor Sets -> Groups
and say that sets are groups with extra properties
(to wit, the property of being free).
OTOH, the Abelianization functor Groups -> Abelian groups
is surjective on the objects (and on the morphisms for that matter),
but groups are not Abelian groups with extra structure,
because the functor isn't injective on the morphisms between a given pair.