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Conjecturally, the theory of motives is essentially determined by intersection theory of schemes, while higher Chow groups (i.e. motivic cohomology) should be to motives what Deligne cohomology is to mixed … At last, the category of pure motives mentioned above should be a full tensor subcategory of the abelian category of mixed motives over the ground field. …

\to A_i$ has a retraction, so that, writing $CHM(E)$ as the $2$-colimit of the categories $CHM(A_i)$, we see easily that the extension of scalars functors is again faithful and conservative (for Chow motives … If you really want Chow motives with integral coefficients, you mays still have to invert the (exponential) characteristic of $F$. …

Let $DM_{gm}(k)$ be Voevodsky's triangulated category of geometric (=constructible) motives over $k$, with rationnal coefficients. … The objects of the category $MM(k)$ are, by definition, mixed motives. …

There is, for each field $k$ the category $M(k)$ of pure motives over $k$, where the Hom' s are Chow groups up to numerical equivalence. … cohomology over $k_i$ for which the standard conjectures hold, then the category $M(k)$ is the filtered 2-colimit of the tannakian categories $M(k_i)$: this comes from the fact this is true for the version of motives …

category $\mathsf{MM}(\mathbf{C})$ is expected to be the heart of the motivic $t$-structure on the triangulated category $\mathsf{DM}_{et}(\mathbf{C})$ of Voevodsky's constructible (or geometric) étale motives … (the one obtained from étale sheaves with transfers; the triangulated category of motives obtained from Nisnevich sheaves with transfers (which is known to coincide with $\mathsf{DM}_{et}(\mathbf{C}) …

Either way, none of these constructions determines Voevodsky's theory of motives. … Finally, this idea of defining motives by forcing Betti cohomology to be a fiber functor is not new at all: if we restrict to pure motives (motives of smooth and projective varieties), this boils down …

The Tate twist is what we need to express Poincaré duality without making any choice. Such a choice appears in the choice of an orientation of the affine line minus the origin, and have shadows in the …

There is no need a priori to define these categories of motives starting from correspondences. … Each of these properties drives us to some theory of of motives. If you insist on finite correspondences, you will go in the direction of $DM(X,\mathbb{Q})$. …