Re: one-object closed categories
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
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.
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
participants (3)
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Mamuka Jibladze -
Peter Selinger -
Tom Leinster