20 - Motivating Motives

Indeed, there is a category $\mathsf{Var}$ of smooth projective varieties and regular maps over $\mathbb{F}_p$, and a functor $$ h \colon \mathsf{Var}^{\text{op}} \to \mathsf{Mot} $$ where $\mathsf{Mot}$, the category of "pure Chow motives", resembles categories we know from linear algebra:

It is a "linear category": the hom-sets are vector spaces, and composition is bilinear.
It is "Cauchy complete", or "Karoubian": it has direct sums, and any $p \colon X \to X$ with $p^2 = p$ is projection onto $Y$ for some direct sum decomposition $X \cong Y \oplus Z$.
It is "symmetric monoidal": it has a well-behaved tensor product $\otimes$, coming from the cartesian product of smooth projective varieties.