  We define an `enriched' notion of Chow groups for algebraic varieties,
agreeing with the conventional notion for complete varieties, but enjoying a
functorial push-forward for arbitrary maps. This tool allows us to glue
intersection-theoretic information across elements of a stratification of a
variety; we illustrate this operation by giving a direct construction of
Chern-Schwartz-MacPherson classes of singular varieties, providing a new proof
of an old (and long since settled) conjecture of Deligne and Grothendieck.
