Grothendieck conjectured that every Weil cohomology theory factors uniquely through some category, which he called the category of motives. For smooth projective varieties (over some field ) such a category was given by Grothendieck himself, called the category of pure Chow motives. For general smooth varieties the category is still conjectural, see at mixed motives.
Fix some adequate equivalence relation (e.g. rational equivalence). Let denote the group of -codimensional algebraic cycles and let denote the quotient .
Category of correspondences
Let , the category of correspondences, be the category whose objects are smooth projective varieties and whose hom-sets are the direct sum
where are the irreducible components of and are their respective dimensions. The composition of two morphisms and is given by
where denotes the projection and so on, and denotes the intersection product in .
There is a canonical contravariant functor
from the category of smooth projective varieties over given by mapping and a morphism to its graph, the image of its graph morphism .
The category of correspondences is symmetric monoidal with .
We also define a category of correspondences with coefficients in some commutative ring , by tensoring the morphisms with ; this is an -linear category additive symmetric monoidal category.
Category of effective pure motives
The Karoubi envelope (pseudo-abelianisation) of is called the category of effective pure motives (with coefficients in and with respect to the equivalence relation ), denoted .
Explicitly its objects are pairs with a smooth projective variety and an idempotent, and morphisms from to are morphisms in of the form with .
This is still a symmetric monoidal category with . Further it is Karoubian, -linear and additive.
The image of under the above functor
is the the motive of .
Category of pure motives
There exists a motive , called the Lefschetz motive, such that the motive of the projective line decomposes as
To get a rigid category we formally invert the Lefschetz motive and get a category
the category of pure motives (with coefficients in and with respect to ).
This is a rigid, Karoubian, symmetric monoidal category. Its objects are triples with .
Category of pure Chow motives
When the relation is rational equivalence then are the Chow groups, and is called the category of pure Chow motives.
Category of pure numerical motives
When the relation is numerical equivalence, then one obtains numerical motives.
Yuri Manin, Correspondences, motifs and monoidal transformations , Math. USSR Sb. 6 439, 1968(pdf, web)
Tony Scholl?, Classical motives, in Motives, Seattle 1991. Proc Symp. Pure Math 55 (1994), part 1, 163-187 (pdf)
James Milne, Motives – Grothendieck’s Dream (pdf)
Minhyong Kim, Classical Motives: Motivic -functions (pdf)
Bruno Kahn, pdf slides on pure motives
R. Sujatha, Motives from a categorical point of view, Lecture notes (2008) (pdf)
Section 8.2 of