Re: one-object closed categories

[Mamuka Jibladze <jib@rmi.acnet.ge>]

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