How to build motivic realizations?
The theory of motives is a grand program envisioned by Grothendieck to encapsulate, within a single framework, the essential features shared by various cohomology theories developed by his school for smooth projective varieties over a field $k$, which are nowadays called Weil cohomology theories. Typical examples include $\ell$-adic cohomology, algebraic de Rham cohomology, and Betti cohomology. The notion of pure motives was introduced by Grothendieck, along with the expectation that there should exist a universal Weil cohomology theory reproducing all known properties of the existing ones. A natural candidate for the category of pure motives is the category of Chow motives introduced by Grothendieck. However, this approach immediately leads to the notorious standard conjectures, which remain unproven to this day. It is also natural to imagine that one can define motives for smooth but possibly non-projective varieties, thereby obtaining the notion of mixed motives $\mathrm{MM}(k)$, a...