A motive is an object designed to encode the common algebraic part of the different cohomology theories of algebraic varieties. In the category of pure motives, morphisms are represented by algebraic correspondences modulo a chosen equivalence relation, and idempotent correspondences are formally split. This construction permits direct sums, tensor products, and duals, while functors to singular, de Rham, or étale cohomology give realizations of a motive.
The theory of mixed motives extends this framework to varieties that are not smooth and projective, but its most general expected category remains conjectural.
