If we have a map p: X --> Y of topological spaces, we can make a definition expressing that the topological type of the fibers of p varies continuously (edit: better to say "locally constantly", thanks Dave) with the base: we can say that p is a fiber bundle.

My question is, can we capture this notion algebro-geometrically, in the case where X and Y are varieties over a field of characteristic zero and p is a map of varieties? I'm looking for a definition hopefully having the following properties (side question: do these seem reasonable?):

1) If X and Y are over the complex numbers, then p is an algebro-geometric fiber bundle if and only if it is a topological fiber bundle on complex points;

2) If f: X-->Y is arbitrary then there is an algebraic stratification of Y such that over each stratum f is a fiber bundle.

Examples should include smooth maps having smooth proper compactifications for which the boundary divisors are in strict normal crossings position, but I would rather the definition not be along these lines because, for instance, I don't want to need resolution of singularities to check that the structure map to the ground field is a fiber bundle.

Edit: In response to several comments, yes, another example would be the normalization map for a cuspidal singularity. In fact I would like the definition to be "topological", in the sense that it factors through h-sheafification.

Edit 2: Whoops, it looks like I used some bad terminology, which probably led to misinterpretations. Sorry folks! To fix things I've replaced all instances of "fibration" with "fiber bundle".

Any thoughts are appreciated!

leftadjoint p_! to p^*, with the proviso that it really lands in pro-objects; then apply it to the terminal object up top and ask if the profinite completion of the thing below is a formal inverse limit of finite locally constant sheaves of homotopy types... but I'd really prefer something more manifestly geometric :) $\endgroup$8more comments