Subject: categories: looking for references

From: Gaunce Lewis <[email-masked]>
Date: Fri, 27 Aug 1999 16:01:06 -0400

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

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From: Ross Street <[email-masked]>
Date: Sat, 28 Aug 1999 13:34:35 +1000

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