In algebraic geometry, a motive (or sometimes motif, following French usage) denotes 'some essential part of an algebraic variety'. One has a satisfactory description of pure motives, whereas the mixed case is much more difficult. Pure motives are triples (X, p, m), where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer. A morphism from (X, p, m) to (Y, q, n) is given by a correspondence of degree n – m.
As far as mixed motives, following Alexander Grothendieck, mathematicians are working to find a suitable definition which will then provide a "universal" cohomology theory. In terms of category theory, it was intended to have a definition via splitting idempotents in a category of algebraic correspondences. The way ahead for that definition has been blocked for some decades by the failure to prove the standard conjectures on algebraic cycles. This prevents the category from having enough morphisms. While the category of motives was supposed to be the universal Weil cohomology much discussed in the years 1960-1970, that hope for it remains unfulfilled. On the other hand, by a quite different route, motivic cohomology now has a technically adequate definition.