John Baez wrote in part:

Ah, the sins of my wayward youth.

Well, Jim is right,
but it turns out that my definition for "structure"
wasn't as bad as we originally thought.
It's wrong -- acording to it, groups aren't sets with extra structure,
simply because the empy set cannot be made into a group --
but it *is* essentially the definition of being *only* extra structure.

First note the theorem that a functor between categories
is an equivalence iff it's full, faithful, and essentially surjective
(that is surjective, not on objects, but on isomorphism classes of objects).
This is analogous to the theorem in set theory that
a function is a bijection iff it's injective and surjective,
as you'll see below.

Now, a forgetful functor is *only* extra stuff iff
it is both essentially surjective and full.
A forgetful functor is *only* extra structure iff
it is both essentially surjective and faithful.
And of course it is (only) extra property iff
it is both faithful and full.
So groups are indeed sets with extra structure,
but they aren't *only* extra structure --
they *also* have the property of being nonempty.

Just as a function between sets can be factored
in a unique way (up to bijection of sets)
into a surjection followed by an injection,
so a functor between categories can be factored
in a unique way (up to equivalence of categories)
into a functor that is only extra stuff,
followed by one that is only extra structure,
followed by one that is extra property.

Also note that while only stuff, only structure, and property
form a complete trio (so long as we stick to 1categories),
stuff, structure, and property is an incomplete list -- downwards.
We can have an arbitrary functor -- that's stuff.
Then we can require that the functor be faithful -- that's structure.
Then we also can require that the functor be full -- that's property.
But there is one more requirement to add of course,
that it be essentially surjective -- that's an equivalence.
So it really goes: equivalence, property, structure, stuff.

To continue with the dimension that Jim forgot at first (-2),
U is an equivalence of categories iff
its homotopy fibres all have dimension -2.

So if U: C -> D is an equivalence of categories,
then an object of D just *is* an object of C
and (once U has been specified) that's all that there is to say about it.
Similarly, there just *is* a -2category,
and that's all that there is to say about it.

But if U: C -> D is full and faithful (extra property),
then given an object of D, it either is or is not an object of C.
The answer to the question is a truth value, a -1category.

Then if U: C -> D is faithful (extra structure),
then given an object of D,
it may be given the structure of being an object of C
in many ways, or one way, or none.
The answer to the question is a set, a 0category.

Then if U: C -> D is any functor whatsoever (extra stuff),
then given an object of D, how many ways can it be given the stuff of C?
The answer to the question is a category.

For example, is a Riemannian manifold,
which is naturally equipped with a connection on the tangent bundle,
a space-with-connection with extra structure
(to wit the structure of a Riemannian metric that reproduces the connection)?
We have discussed this on this board before.

This is, in part, because isometries preserve the connection.

This is because uninvertible unitary maps need not preserve the connection,
so there is no functor.

The problem here has light shed on it by the following example:

A _semigroup_ is a set with a binary operation;
a _monoid_ is a semigroup with an identity element.
(A group is a monoid with inverses.)
Now, is a monoid a semigroup with extra property?
The definition as phrased above suggests so,
and from the groupoid POV that is entirely correct.
But the functor from _Mon_ to _SGrp_ is not full on *all* morphisms,
because monoid homomorphisms are required to preserve the identity.
Thus, from the *morphisms'* POV, the identity is a structure to be preserved.
Had we not made this requirement on monoid homomorphisms,
then a monoid would indeed be a semigroup with extra property, from any POV.

I like to describe the situation in which
the groupoid POV says that the functor is extra property
but the category POV says only that the functor is extra structure
by saying that there *is* extra structure
but the structure is *definable* in terms of the old structure
(in this case, the structure of the semigroup alone, the binary operation).
Jim doesn't like this way of looking at it,
but we do agree that the phenomenon is there
(and has something to do with definability in a logical sense).

Now we see what went wrong with
the functor from _RiemMan_ to _ManW/Conn_,
the functor that was extra structure from the groupoid POV
but didn't even exist from the category POV.
The problem is that the morphisms in _RiemMan_
did not preserve the Riemannian connection.
We can create a new category, _RiemManW/Conn_,
whose morphisms *are* required to preserve this
(in addition to the metric itself).
Then the picture looks like this:

   RiemManW/Conn
      /     \
     /       \
RiemMan    ManW/Conn

The functor on the right is the functor of extra structure, from any POV.
The functor on the left is *also* a functor of extra structure,
in fact a functor of *only* extra structure, from the category POV.
However, this extra structure (the Levi Civita connection)
is definable in terms of the structure in _RiemMan_ (the metric).
So from the groupoid POV, it's actually a functor of extra property.
However, there is no extra property in this structure --
*only* extra structure, because every metric has a Levi Civita connection --
so in fact the extra property from the groupoid POV is the vacuous property
and the functor on the left is an equivalence of groupoids.
This makes it invertible, so -- from the groupoid POV --
the bottom row can be filled in with a functor from _RiemMan_ to _ManW/Conn_
that, like the functor on the right, is of extra structure.
But from the category POV, no such functor exists.

All of this -n-category stuff suggests a creation story that I'm developing.

In the beginning, there was nothing.
But with nothing came nothingness, the vacuity, which was something.
So now there was nothingness and somethingness,
vacuity and triviality, falsehood and truth,
which were 2 things.
So now there was a set of nothingness and somethingness,
and the whole realm of sets of elements
sprang up out of the set of nothingness and somethingness,
and sprouted functions between them
to relate them back to the set of nothingness and somethingness.
So now there was a category of sets and functions,
and the whole realm of categories of objects and morphisms
sprang up out of the category of sets and functions,
and sprouted functors between them,
to relate them back to the category of sets and functions,
which sprouted natural transformations between *them*,
to relate the relationships.
So now there was ...

-- Toby
   t...@math.ucr.edu