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