Fortifying the Yomdin-Gromov Algebraic Lemma

We provide sharp cylindrical parametrizations of cylindrical cell decompositions by maps with bounded $C^{r}$ norm in the sharply o-minimal setting, thus generalizing and strengthening the Yomdin-Gromov Algebraic Lemma. We introduce forts, geometrical objects encoding the combinatorial structure of cylindrical cell decompositions in o-minimal geometry. Cylindical decompositions, refinements of such decompositions, and cylindrical parametrizations of such decomposition become morphisms in the category of forts. We formulate and prove the above results in the language of forts.

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