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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]