Cat as a '2-fibration' over Set

[David Roberts <droberts@maths.adelaide.edu.au>]

Hi all,

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.
This can be easily seen by constructing, given a category C = (C_1
\rightrightarrows C_0) and a function f:A \to C_0, the set of arrows
A^2 \times_{f,C_0^2}C_1 (the pullback of (s,t):C_1 \to C_0^2) of the
category C[f].

The cartesian lift of f is then the canonical functor F:C[f]\to C.

Now given another function g:A\to C_0 -- giving rise to G:C[g]\to C --
and a natural transformation F \Rightarrow G there is a canonical
isomorphism C[f]\simeq C[g] over C. Thus if we think of Cat as a
2-category, there is something extra going on. For example, one gets a
pseudofunctor Set \to 2Cat on choosing specified pullbacks to define
C[f].

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?

David


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