In mathematics, a bivariant theory was introduced by Fulton and MacPherson (Fulton & MacPherson 1981), in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring.
Intuitively, the consideration of all the fiber squares like the above can be thought of as an approximation of the map
is a family of group homomorphisms indexed by the fiber squares: satisfying the certain compatibility conditions.
(Totaro 2014) has shown that rationally there is no such a map with good properties even if X is a linear variety, roughly a variety admitting a cell decomposition.
He also notes that Voevodsky's motivic cohomology ring is "probably more useful " than the operational Chow ring for a singular scheme (§ 8 of loc.