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!



Definition of fiber bundle in algebraic geometry
 ~~OK, let me venture to give a definition. Say that a morphism $f:X\to Y$, of varieties over
a field, is an algebraic fibration if there exists a factorization
$X\to \overline{X}\to Y$, such that that the first map is an open immersion, and the
second map is proper and there exists a partition into Zariski locally closed strata $\overline{X}=\coprod\overline{X}\_i$, such that restrictions 
$\overline{X}\_i\to Y$ are smooth and proper. $X$ should be a union of strata.
Perhaps, one should also insist that this is Whitney stratification.~~ 




Shenghao's comment made me realize that my original attempt at an answer was problematic.
Rather than trying to fix it, let me make a fresh start. 
Let us say that $f:X\to Y$ is an algebraic fibration if there exist a simplicial scheme $\bar X\_\bullet$ with a divisor $D\_\bullet\subset \bar X\_\bullet$ such that


1. There is map $\bar X\_\bullet -D\_\bullet\to X$ satisfying cohomological descent, in the
sense of Hodge III, for the classical topology (over $\mathbb{C}$) or etale topology (in general).
2. The composite $\bar X\_n\to Y$ is smooth and proper, and $D\_n$ has relative normal crossings for each $n$.


These conditions will ensure that $R^if\_\*\mathbb{Z}$ (resp. $R^if\_\*\mathbb{Z}/\ell \mathbb{Z}$)
are locally constant etc.


I think that this would also apply Shenghao's question 


[What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?](https://mathoverflow.net/questions/72446/what-would-be-a-characteristic-p-analogue-for-c-infty-fiber-bundles)


Although I won't claim that this is in any sense optimal.


Oh, and I forgot to say that when $Y=Spec k$ is a point, every $X$ can be seen to be fibration (as it should) by De Jong




