In article <72b3pd$...@gap.cco.caltech.edu>,

Of course, part of the point of this puzzle is that the term
"forgetful" is usually given no precise definition, so here we
are seeking a precise definition.  Usually people don't bother
to define "forgetful functor" very precisely - like pornography,
you're just supposed to know a forgetful functor when you see it.

I'm a little unhappy with this for two reasons: one nitpicky and one
more serious.  

The nitpicky reason is that it's bad to care if a functor U: C -> D
is surjective on objects.  If you think you want this, all you *really*
should want is that U be "essentially surjective".  This means that
every object of D is, not necessarily equal, but *isomorphic* to an
object of the form U(x) for some object x of C.  

In general, the interesting properties of functors must be preserved
by natural isomorphisms.  If you have two naturally isomorphic functors
and one is essentially surjective, so is the other.  This is not true
of "surjective on objects".  

However, the more serious reason I'm unhappy is that sometimes C-objects
are D-objects with extra structure but *not every D-object can be made
into a C-object*.  In this case U: C -> D is not essentially surjective.

For example, consider the forgetful functor U: Vect -> Set where Vect
is the set of real vector spaces.  There's no way to equip a set with
2 elements with the structure of a real vector space!  Thus U is not
essentially surjective.  

Anyway, I think we should say that C-objects are D-objects with
extra structure if your second criterion holds: given any pair of
objects x,y in C,

U: hom(x,y) -> hom(Ux,Uy)

is injective.  By the way, a functor with this property is called
"faithful".  

Hmm, again I'm unhappy for the same sort of nitpicky reason.  Again,
it's bad to care if U is injective on objects, because this property
is not preserved by natural isomorphisms.  I believe the politically
correct substitute for this property is called "reflecting isomorphisms":
we say a functor U: C -> D "reflects isomorphisms" if U(f) being an
isomorphism in D implies that f is an isomorphism in C.   In particular,
nonisomorphic objects in C can't get sent to isomorphic objects in D
by a functor that reflects isomorphisms.

It seems that whenever C-objects are D-objects with extra properties,
the forgetful functor U: C -> D reflects isomorphisms.  For example,
if two groups are isomorphic, and they happen to be abelian,
they are isomorphic in the category of abelian groups.

But I'd give a slightly different criterion for when C-objects are
D-objects with extra properties.  I'd say this happens when U: C -> D
is "fully faithful".  This means that given any pair of objects
x,y in C,

U: hom(x,y) -> hom(Ux,Uy)

is 1-1 and onto.   (If this map is always injective, we say it's
"faithful".  If it's always surjective, we say it's "full".  If
both, we say it's "full and faithful", or "fully faithful" for
short.)

Note that a fully faithful functor always reflects isomorphisms -
this is a fun little exercise - so my criterion is stronger than
yours, at least modulo political correctness, which forbids me
from saying that a functor is "injective on objects".

Also note that the way I'm setting things up, "extra properties"
is a special case of "extra structure".  

Hmm, I wouldn't demand that.  An example of "extra stuff" would
be the functor U: Vect^2 -> Vect that takes a pair of vector spaces,
or a pair of linear map, and discards the second one.  In other words,
I'd say a pair of vector spaces is a vector space "with extra stuff",
namely another vector space.  The functor U: Vect^2 -> Vect doesn't
have the property you demand - i.e., it's not faithful.  

Now I forget if there is *any* property we should demand of U: C -> D
when D-objects are supposed to be C-objects with "extra stuff"!
Maybe James Dolan will remind me what he told me about this case.

Let's check these examples: yes, U: Groups -> Sets is
faithful, while U: Abelian Groups -> Sets is fully faithful.

Hmm, F: Sets -> Groups is faithful, but not full.  Thus I'd
say that a set can be viewed as a group with extra *structure*:
namely, the property of being free together with the *structure*
of a specific set of generators.  The point is that not all
homomorphisms between free groups come from functions between
their set of generators.

Fun stuff, eh?   But I'm afraid it's drifting rather far afield
from physics, except insofar as every mathematical physicist
should spend a little time thinking about "properties" vs
"structure".