Q:

Axiomatic intersection theory

Is there an axiomatic intersection theory?
What I expect is something like:
An intersection theory is a functor from the category of schemes(or other spaces) to the category of algebras, with well-defined flat pullback and proper pushforward, and (maybe) projection formula or some other axioms.

A:

You might be interested in the bivariant theories of Fulton and MacPherson.

