What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences? 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?

 A: There is no need a priori to define these categories of motives starting from correspondences. The stable homotopy theory of schemes $SH$ may be characterized by a universal property saying that it is the universal setting in which one may define the six operations; see Drew and Gallauer. Usually, if you have a system of coefficients $D$ in which the six operations are defined, we have in particular: for each morphism of scheme $f:X\to Y$, a pull-back functor $f^*:D(Y)\to D(X)$ with a right adjoint $f_*$ as well as a push-forward with compact support functor $f_!:D(X)\to D(Y)$ with right adjoint $f^!$ satisfying a bunch of properties, among which we ask for homotopy invariance (i.e. full faithfulness of $f^*$ in the case where $f$ is the structural map of a vector bundle). The universal property of $SH$ means that $SH$ has the six operations and is initial for this, as far as we ask that $D$ takes values in presentable stable $\infty$-categories. More precisely, the assignment $(f:Y\to X)\mapsto f_!f^!(1_X)$ defines a functor from smooth $X$-schemes to $D(X)$ for each $X$, and this extends uniquely into a compatible family of functors $SH(X)\to D(X)$. For example, we may take $D(X)=D(Sh(X_{\acute{e}t},\Lambda))$ with $\Lambda$ any ring of positive characteistic invertible $\mathcal{O}_X$. OR we may consider its $\ell$-adic counterpart $D(X,\mathbb{Z}_{\ell}):=D(Sh(X_{\acute{e}t},\mathbb{Z}_{(\ell)}))/D(Sh(X_{\acute{e}t},\mathbb{Q}))$ with $\frac{1}{\ell}\in\mathcal{O}_X$.
If we focus on $\mathbb{Q}$ coefficients, we may consider the $\mathbb{Q}$-linear version of $SH$, and consider one of the following properties:

*

*we want representable cohomology theories to be graded in the usual sense (because, due to the interpretation of quadratic forms in $SH$, there is an element $\varepsilon$ which is a non-trivial square root of $1$ but which is not always equal to $-1$).

*we want the projective bundle formula to hold (equivalently, we want Chern classes to exist in our cohomology theories).

*we want étale descent (or hyperdescent).

*we want étale descent and proper descent (or hyperdescent).

*we want (finite) correspondences to act on cohomology theories.

It happens that one of these properties hold in a $\mathbb{Q}$-linear $D$ if and only if all of them do. Or, equivalently, if we force one of these properties to hold in the $\mathbb{Q}$-linear version of $SH$, all the other ones hold as well. 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 $DM(X,\mathbb{Q})$. If you insist on the graded thing, you will consider Morel's $SH(X)_{\mathbb{Q},+}$. If you prefer étale descent, you get what Ayoub likes to call $DA(X,\mathbb{Q})$. If you insist on proper descent, you get Voevodsky's $DM_h(X,\mathbb{Q})$, constructed from $h$-sheaves. One way to make Chern classes appear naturally is to use representability algebraic $K$-theory in $SH$, and this leads to we are called Beilinson motives here.
But, as shown in loc. cit., all these constructions are equivalent (except for $DM(X,\mathbb{Q})$ where we also need $X$ to be geometrically unibranch because finite correspondences are tricky to control over non-normal schemes), and many of them do not mention correspondences in their definition. And these theories are nice because Chow groups naturally are representable in there, so that we get the lovely picture that classical intersection theory has something to do with the universal cohomology theory (this is a super fancy and efficient way to construct cycle classes).
The reason why you may see people focusing on correspondences is because they appear naturally via Poincaré duality (Gysin maps), but also mainly because they want to be able to compute: the construction of motivic categories eventually demand that we invert the Tate object, and that leads to categories which are far from usual derived categories, thus making concrete computations easily out of reach. In versions with correspondences, and if we restrict ourselves to motives over a field, the constructions that involve finite correspondences usually imply that the inversion of the Tate object is less dramatic, so that we can still control what is happening. This kind of results are called cancellation theorems and usually are technical (I mean that the proof requires non-formal arguments which only hold over fields and that such cancellation results are known to be false over higher dimensional base schemes). The cancellation theorem for $DM$ is proved by Voevodsky, which implies the full faithfulness of the functor $DM^{eff}_{Nis}(k)\to DM(k)$. Another reason to go through correspondences is that, with integral coefficients, having a nice theory of Chern classes is not a mere property, but a genuine structure, and making this precise in a coherent way is challenging (especially if we do not work in a context with étale descent). For a conceptual approach to correspondences in $SH$ (for $R$-modules in $SH$ with $R$ a motivic ring spectrum), I would recommend to have a look at the work of Elden Elmanto, Marc Hoyois, Adeel A. Khan, Vladimir Sosnilo, and Maria Yakerson. The fact that one can prove a cancellation theorem with an appropriate notion of finite correspondences for $SH$ is also due to Voevodsky and has been developed in full by the aforementioned authors as well as by Alexey Ananyevskiy, Grigory Garkusha and Ivan Panin.
