7
$\begingroup$

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 $C[f]$.

My question(s):

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?

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.