In
algebraic geometry, a
motive (or sometimes
motif, following
French usage) denotes 'some essential part of an
algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. 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.