Re: one-object closed categories

[Tom Leinster <T.Leinster@dpmms.cam.ac.uk>]

> From: maxk@maths.usyd.edu.au (Max Kelly)
> 
> Tom observed that an abelian monoid is a symmetric monoidal closed
> category with one object, and asked whether anyone had studied categories
> enriched in such a closed category.
> 
 [...]
> 
> Anyway, I had a brief look at V-categories for such a V tonight, but with
> too few details so far to say much about them before bedtime. Queer little
> creatures, aren't they? A V-category A has objects a, b. c. and so on, but
> each A(a,b) is the unique object * of V. All the action takes place at the
> level of j: I --> A(a,a) and M: A(b,c) o A(a,b) --> A(a,c).
 [...]

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. 

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.

Tom