Re: one-object closed categories

Re: one-object closed categories

[Tom Leinster]

> 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

Re: Re: one-object closed categories

[Peter Selinger]

> 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.

Re: one-object closed categories

[Mamuka Jibladze]

Concerning categories enriched in monoidal categories with a single
object: another example is given by cocycles. It can be presented in
various ways. For example, a "\v Cech style" version: given an open cover
of X by Ui and a (suitably normalized) \v Cech 3-cocycle c of this cover
with values in M, this enrichs in the evident way the category whose
objects are the i's, with hom(i,j) either a singleton or empty according
to inhabitedness of the intersection of Ui and Uj. Other variations
suggest things like morphisms of simplicial sets to the nerve of M
considered as a 2-category with a single 1-cell. This is related to
K(M,2)-torsors, etc. Quite probably there are several publications
exploiting this. At least cocycles with values in monoids rather than
groups certainly have been considered. 

What I certainly have not seen is a backwards generalization: has anybody
considered analogs of K(M,2)-torsors for general enrichments? Would be
very interested in a reference.

Happy holidays to all!
Mamuka