Re: Cat as a '2-fibration' over Set

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

On 07/10/2010, at 8:18 PM, David Roberts wrote:

> To start with think of Cat as a 1-category. The functor Obj:Cat \to
> Set sending a small category to its set of objects is a fibration.


Dear David

In a daring version of an undergraduate algebra unit on groups,
I taught the notions of cartesian and opcartesian morphism
for a functor and looked at them for the functor ob : Cat --> Set.
The goal was to give a groupoid proof of the Nielsen-Schreier
theorem using fibrations in the small (between groupoids) and in the
large. I achieved the goal to my own satisfaction; I think most of
the students thought otherwise. A core of them liked it. This is
the most explicit category theory I have tried to teach pre fourth
year honours.

My inspiration very definitely came from Ronnie Brown's topology  
book(s).

I'm not at work today (Saturday, and a grandson's birthday party)
so I can't check whether these constructions of direct and inverse
images for ob : Cat --> Set are in that book, whether it is the
ob : Gpd --> Set  case that is there, or what. Ronnie can tell us
perhaps. Anyway, it is essentially there. It may not be phrased in
terms of cartesian morphisms.

> Has this phenomenon been studied before? (I would think so)
> Does this make Obj a fibration of 2-categories (see e.g. Hermida, or  
> Bakovic)?
> Or is this a more 'classical' concept? More basically, where was this
> fact first pointed out?

I too would like to know of other references.

I am ashamed to say I hadn't thought about the 2-fibrational aspects
of ob : Cat --> Set.

Also, how about the Beck-Bénabou-Roubaud-Chevalley condition?

Ross



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