Re: mystification and categorification

[Tom Leinster <tl@maths.gla.ac.uk>]

Steve Schanuel wrote:
> a category with both plus and times as extra monoidal structures.
> (Does anyone know an example of interest in which the plus is not
> coproduct?)

Here are two examples that I've come across previously of rig categories
in which the plus is not coproduct:

(i) the category of finite sets and bijections, with + and x inherited
from the category of sets;

(ii) discrete rig categories, which are of course the same thing as
rigs.

> This freedom is unnecessary; a little algebra plus Robbie
> Gates' theorem provides a solution G to  G^2=G+1 which satisfies no
> additional equations, in an extensive category (with coproduct as plus,
> cartesian product as times).

If you *do* allow yourself the freedom to use any rig category then an
even simpler solution exists, also satisfying no additional equations:
just take the rig freely generated by an element G satisfying G^2 = G +
1 and regard it as a discrete rig category.

>     Since in the category of sets, any nasty old infinite set satisfies
> the golden equation and many others, I have taken the liberty of
> interpreting  'nice' to mean at least 'satisfying no unexpected
> equations'.

I'd interpret "nice" differently.  (Apart from anything else, the
trivial example in my previous paragraph would otherwise make the golden
object problem uninteresting.)  "Nice" as I understand it is not a
precise term - at least, I don't know how to make it precise.  Maybe I
can explain my interpretation by analogy with the equation T = 1 + T^2.
A nice solution T would be the set of finite, binary, planar trees
together with the usual isomorphism T -~-> 1 + T^2; a nasty solution
would be a random infinite set T with a random isomorphism to 1 + T^2.
(Both these solutions are in the rig category Set with its standard +
and x.)  I regard the first solution as nice because I can see some
combinatorial content to it (and maybe, at the back of my mind, because
it has a constructive feel), and the second as nasty because I can't.
I'm not certain what I think of the solution given by the set of
not-necessarily-finite binary planar trees (nice?), or by the set of
binary planar trees of cardinality at most aleph_5 (probably nasty).

Maybe the finding of a "nice" solution is similar in spirit to the
finding of a "concrete interpretation" of a combinatorial identity.  As
an extremely simple example, consider the identity saying that each
entry in Pascal's triangle is the sum of the two above it,

   (n+1 choose r) = (n choose r-1) + (n choose r).

This is a doddle to prove, but all the same you'd be missing something
if you didn't know the standard "concrete interpretation": choosing r
objects out of n+1 objects amounts to EITHER choosing the first one and
then choosing r-1 of the remaining n OR ... .  Even if the challenge of
finding a "nice solution" or "concrete interpretation" isn't made
precise, I think there is a shared sense of what would count as an
answer, and finding an answer is in general not straightforward.

Best wishes,
Tom