Re: Re: one-object closed categories

[Peter Selinger <selinger@math.lsa.umich.edu>]

> From Tom Leinster:
> 
> Since I asked the question I've found a few examples; they've all got
> the same flavour about them, so I'll just do my favourite.
> 
> If V is the commutative monoid, then a V-enriched category is a set A plus
> two functions
> 	[-,-,-]: A x A x A ---> V
> 	    [-]:         A ---> V
> satisfying
> 	[a,c,d] + [a,b,c] = [a,b,d] + [b,c,d]
> 	  [a,a,b] + [a] = 0 = [a,b,b] + [b]
> for all a, b, c, d. 

A few remarks: In the case where V is an abelian group, the first
axiom already implies the other two if we define [a] = -[a,a,a].
Namely, by letting a=b in the first axiom, it follows that [a,a,c] is
independent of c.

If V is an abelian group, then one can get an example of the above
structure from an arbitrary map {-,-} : A x A ---> V by letting
[a,b,c] = {a,b}+{b,c}-{a,c} and [a] = -{a,a}. Tom's "area" example is
of this form.

In fact, if V is an abelian group, then *any* example of a V-enriched
category is (non-uniquely) of the form described in the previous
paragraph: Fix some x in A (if any), and define {a,b} = [a,b,x].

What about the non-group case? In general, [a,b,c] need not always be
invertible in V. In fact, [a,b,a] need not be invertible. For a simple
example of this, let V be the natural numbers and define

 [a]     = 0,
 [a,b,c] = 0, if a=b or b=c,
           1, if a,b,c pairwise distinct,
           2, otherwise (i.e., if a=c but a,b distinct).

This indeed works. 

Best wishes, 
-- Peter Selinger

> The example: let A be a subset of the plane. Choose a smooth path P(a,b) from
> a to b for each (a,b) in A x A, and define [a,b,c] to be the signed area
> bounded by the loop
> 	P(a,b) then P(b,c) then (P(a,c) run backwards);
> also define [a] to be 
> 	-(area bounded by P(a,a)). 
> (There's meant to be an orientation on the plane, so that areas can be
> negative.)  Then the equations say obvious things about area - don't think
> I'm up to that kind of ASCII art, though.