I come today to mathoverflow to showcase some genuine confusion about the motivic world. I want to ask some questions before actually starting to study the subject, to build some sense of direction.

I take Grothedieck's orginal idea of motives to be that of an abelian category through which every good cohomology should factorize.

Skimming through the Morel-Voevodskij paper, $\mathbb{A}^1$-homotopy theory is what you get when you: start with the category of smooth $k$-schemes of finite type; take Nisnevich topology on it; take simplicial sheaves for this topology; put a model structure where weak equivalences are stalkwise; finally localize to make the projections $\mathbb{A}^1 \times{X} \to X$ weak equivalences. I will call this category $\text{MV}_k,$ and its homotopy category $\text{H}(k).$

I also know from here that the model category defined above is Quillen equivalent to some localization (imposing sheaf conditions and $\mathbb{A}^1 \times{X} \xrightarrow{\sim} X$) of the category of all simplicial presheaves on smooth schemes of finite type with the projective model structure; this latter category enjoys a universal property being a localization of the universal model category on smooth schemes of finite type over $k.$

What's the relationship of this category with motives?

Skimming through the motivic cohomology book, I see that the triangulated category of motives is defined in the following way: first one takes the additive category of correspondences $\text{Corr}_k,$ then takes $\text{Ab}$-enriched preasheaves on $\text{Corr}_k^{\text{op}},$ then takes sheaves for the Nisnevich topology, takes the derived category of this, and then localizes at $\mathbb{A}^1$-weak equivalences. This category I guess would be denoted $\text{DM}^{\text{eff}}_{\text{Nis}}(k,\mathbb{Z}).$ For some general commutative ring $R,$ one takes instead $\text{Ab}$-enriched presheaves and sheaves of $R$-modules to define $\text{DM}_{\text{Nis}}^{\text{eff}}(k,R).$ I hope I got this right.

What is the relationship between this latter construction and the first one?

I also would like to understand better the intuition behind the use of the category $\text{Corr}_k$.

An elementary correspondence between $X$ and $Y$ is an irreducible closed subset of $X \times{Y}.$ In $\text{Corr}_k$ objects are smooth separated schemes of finite type and the set of morphisms between $X$ and $Y$ is the free abelian group on elementary correspondences.

So we are choosing certain set of spans between two objects and then taking the free abelian group on it. I also understand that considering elementary correspondences is a way to enlarge the category $Sm_k$ in a such a way that every morphism $f:X \to Y$ is sent to its graph.

I see that correspondences are used in the original formulation of Grothedieck's pure motives. So they were there from the very first idea of motives.

How should I think of correspondences? What is the connection between the idea of correspondence and Grothendieck's original idea of motives?

Why, if the goal is to build a(n abelian) category through which good cohomology theories factorize, we begin the construction starting with correspondences?

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