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][1], $\mathbb{A}^1$-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][2] 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][3], 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? 





  [1]: http://archive.numdam.org/article/PMIHES_1999__90__45_0.pdf
  [2]: https://arxiv.org/pdf/math/0007070.pdf
  [3]: http://www.claymath.org/library/monographs/cmim02.pdf