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We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer–Weiss–Williams family index theorem for the parametrized   A  ‐theory Euler characteristic of a smooth bundle.

Parametrized cobordism categories and the Dwyer–Weiss–Williams index theorem