# What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?

I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the morphism to be smooth; e.g. a family of nodal curves $X_t$ with $p_a=2$ and such that the $j$-invariant of the normalizations $\widetilde{X}_t$ is not constant in $t,$ would certainly qualify.

Motivation: If $f:X\to Y$ is a proper morphism of complex algebraic varieties, then by Morse theory, there exists a "stratification of $f";$ in particular, over each stratum of $Y,\ f$ is a $C^{\infty}$-fiber bundle. I wonder if this could be true in char. $p,$ but the first thing is to have an analogous notion.

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Hey Shenghao, maybe check out this MO question... mathoverflow.net/questions/49759/… –  Kevin H. Lin Aug 9 '11 at 9:25
Thanks, Kevin. I had a question about the suggested definition, and have just posted it there. –  shenghao Aug 9 '11 at 12:09