Re: Terminology of locally small categories without replacement

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

Dear Jean

Thank you for your message. I have been in Canberra and am only now
catching up with emails, reference writing and the like.
In the meantime, I see that you provided an answer to the original
question under this "Subject" (involving locally small fibrations) and
that Richard Garner has responded well to your questions showing the
close relationship between the concepts. I think the comma object
observation (I would not claim it competitively as "my notion"
particularly) was a helpful viewpoint for some researchers. As Richard
points out that, to make the definition, the 2-category K does not
really need any limits however comma objects are useful (in the same
way that internal category can be defined in any category but having
pullbacks makes one feel better). (I now notice in my message that C
somehow was mistyped later as R. As indicated, more conditions on C
give stronger consequences.)

I would be pleased to hear what you have in mind as some of the
significant and numerous results derivable using locally small
fibrations.

Walters and I were interested at one time in developing category
theory in a 2-category with an analogue P of the presheaf construction
("Yoneda structures"). Mark Weber has recently been able to make use
of some of these ideas in developing foundations for recent advances
in Batanin's operad theory. I believe my paper [The petit topos of
globular sets, J. Pure Appl. Algebra 154 (2000) 299-315] was some help
in this respect.

In any case, one example of a 2-category K is provided by a finitely
complete cartesian closed category E with an internal full subcategory
S; we take K = Cat(E) and Pa = [a opposite,S] where [ , ] is cartesian
internal hom in Cat(E).

In particular, we can take E = Cat so that K is the 2-category Dbl of
double categories. Given any internal full subcategory set of Set,
there is an internal subcategory fun of Cat which is the double
category of squares based on categories in set. The objects of fun are
categories in set, the horizontal and vertical morphisms are functors,
and the squares are natural transformations in the squares. The small
objects of K = Dbl are defined to be the double categories in set.

Best wishes,
Ross


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