categories: Pseudo orbits

[Ross Street <street@ics.mq.edu.au>]

Dear Gaunce Lewis, GT-colleague and all,

When we regard a monoid  M  as a one-object category, an M-set is a functor
X : M --> Set  and the colimit of the functor  X  is the set of orbits of
the M-set.

What GT-colleague has is an ordered monoid which can be regarded as a
one-object 2-category  M,  and the action  F  of  M  on the category  C
amounts to a 2-functor X : M --> Cat.  I suspect that the construction
required is the pseudocolimit of  X.  This kind of colimit for 2-functors
was considered in the book of John Gray

J.W. Gray, Formal Category Theory: Adjointness for 2-Categories,  Lecture
Notes in Math. 391 (Springer, 1974)

and in my paper

Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8
(1976) 149-181

where I show that pseudo(co)limits and lax (co)limits are ordinary weighted
(= indexed) (co)limits in the sense of enriched category theory (for the
base monoidal category  Cat).

I suspect the condition that the identity element is initial is a red
herring even though this makes it look as though canonical maps are being
inverted rather than isomorphisms being introduced.

Regards,
Ross