looking for references

[Gaunce Lewis <lglewis@syr.edu>]

A colleague in geometric topology has encountered a categorical 
construction for which he would like some literature references.  He has 
asked me to pass this request on to this mailing list.  Roughly speaking, 
the construction takes a category carrying an action by a monoid and forms 
an associated "orbit" category.  However, rather than identifying objects 
in the same orbit, it inserts a canonical isomorphism between them.

Here are the details:

Let M be a monoid which is also a poset.  Assume that the multiplication on 
M preserves the order and that the unit u for the multiplication on M is an 
initial element for the poset.  Think of M as a category with morphisms 
derived from the poset structure.  Let C be any category and let F : M x C 
-> C be a functor which gives an action of M on C.  For each m in M and 
each c in C, there is a map t(m,c) from c to F(m,c) obtained by applying F 
to the poset relation u \leq m and the identity map on c.  Form the 
category of fractions of C in which all the maps t(m,c) have been 
inverted.  Note that it looks somewhat like the orbit category C/M, but 
with the objects in the same orbit linked by canonical isomorphisms 
(derived from the t(m,c)) rather than identified.

Has anyone seen this construction before?  Is there literature on it?

Thanks for any help on this,
Gaunce Lewis