This question is heavily related to this question.
Fix a sufficiently nice and connected topological space $B$ and let $FB$ be the category of fiber bundles over $B$. A morphism $f: (E\to B)\to (E'\to B)$ in this category is a map $E\to E'$ over $B$ and hence it maps fibers to fibers.
Edit: A fiber bundle $p:E\to B$ is a continuous map such that there exists a local trivialization. Since $B$ is nice and connected, all the fibers $p^{-1}(x)$ are isomorphic.
According to the very helpful answers here,
- a fiber bundle over $B$ with fiber $F$ determines a $G=Aut(F)$-principal bundle together with (trivially) a left $G$-action on $F$ and
- a $G$-principal bundle over $B$, where $G$ is an arbitrary topological group, together with a topological space $F$ and a left $G$-action on $F$ determines a fiber bundle over $B$ with fiber $F$.
Can one formulate this correspondence "categorically", i.e. is there an equivalence of the category $FB$ to a product (?) of two categories, one encoding the "glueing structure" (the principal bundle) and one encoding the "fiber information" (the space $F$ with the action)? (In particular, what should be the analogue to a morphism of fiber bundles?)
(Certainly is not possible to get such a description precisely like above because one have to fix the topological group $G$ to say what a space $F$ with a $G$-action is and conversely one have to fix a space $F$ to say what a $G=Aut(F)$-principal bundle is, but maybe there is a way to formulate this in general.)
Best Answer
It is definitely not an easy read, but you might want to have a look at Peter May's Classifying Spaces and Fibrations monograph. If i recall correctly, he constructs classifying spaces for fibrations with fiber F in some very categorical fashion. So it seems like it might imply something like the result you are interested in, although i doubt it is exactly what you want. It has been a long time since i read through any of it (I never got through all of it). This seems like something you might want to look at, let me know if you are having a hard time finding a copy via email.
Also... I could be completely wrong.