I have some vague knowledge about the philosophy that schemes should be thought of as similar to topologic spaces, and we should divide everything by homotopy, and that the space should be actually sheaf in a correct topology.

Could somebody provide a concise and modern introduction that would allow me to work with statements like (from an answer to question about motivic cohomology)

K(Z(0),0) is simply the constant sheaf Z.


Let $E(S)$ be the category of Nisnevich sheaves on the site of smooth schemes over some base $S$. Then Morel and Voevosy's homotopy category $\mathrm{H}(S)$ is obtained as a localization of the category $sE(S)$ of simplicial objects in $E(S)$. The localization functor $\mathrm{loc}:sE(S)\to \mathrm{H}(S)$ can always by constructed in a way that it is the identity on objects, so that you can always think of the objects as genuine simplicial sheaves. A statement as above reads simply as: the object $K(\mathbb Z(0),0)$ is isomorphic to the image by $\mathrm{loc}$ of the constant sheaf $Z$. Now, if you want to use such a statement to compute something, you will need to know what are the maps in $\mathrm{H}(S)$. For this, you will need quite a bit of general homotopy theory (here the homotopy theory of simplicial sheaves as well as the theory of left Bousfield localization), and, of course, at some point, some geometry. However, the theory of Bousfield localization applies here in a rather gentle way, if you admit the homotopy theory of simplicial sheaves.

Let $\mathrm{Ho}(sE(s))$ be the localization of $sE(S)$ by the class of local weak equivalences (i.e. maps inducing weak equivalences of simplicial sets stackwise). Then, for any smooth $S$-scheme $X$, you have a derived global section functor $R\Gamma(X,?)$ (with values in Kan complexes). Morel and Voevodsky's homotopy category $\mathrm H(S)$ is obtained by inverting some maps in $\mathrm{Ho}(sE(S))$, so that we have a localization functor $\mathrm{loc}:\mathrm{Ho}(sE(S))\to H(S)$. As this comes from a left Bousfield localization at the level of the underlying model categories, this latter localization functor has a right adjoint $i:\mathrm H(S)\to\mathrm{Ho}(sE(S))$ which is fully faithful (almost by construction/definiton). Hence, we can understand $\mathrm H(S)$ as the full subcategory of $\mathrm H(sE(S))$. Moreover, we can understand the essential image of $i$ in a rather simple way: it consists of the objects $F$ such that, for any smooth $S$-scheme $X$, the map $R\Gamma(X,F)\to R\Gamma(X\times \mathbb A^1,F)$ is a weak equivalence of Kan complexes. This means that, whenever you have your favourite cohomology theory $F$, if it satisfies Nisnevich descent (hence is representable in $\mathrm Ho(sE(S))$, then it is representable in $\mathrm H(S)$ if and only if it is homotopy invariant. If it only satisfies Nisnevich descent, then you still have a universal way to force $\mathbb A^1$-homotopy invariance (by applying the functor loc). for instance, $K(\mathbb Z(0),0)$ is really the Nisnevich cohomology with coefficients in $\mathbb Z$. However, in the latter case, you might have some trouble to compute what you get. For the higher $K(\mathbb Z(n),2n)$, there is an explicit description in terms of complexes which is obtained as follows. Let $F$ be a sheaf of abelian groups. Consider the cosimplicial scheme $\Delta^n$; defined as the spectrum of the (sheaf of) ring(s) $\mathcal O[t_0,...,t_n]$ modulo the relation $t_0+...+t_n=1$ ($\mathcal O$ is the sheaf of functions on S). Taking the internal Hom's, you get a simplicial sheaf of abelian groups $\mathrm{Hom}(\Delta^n,F)$ (letting $n$ vary; you can also play with the Dold-Kan correspondance to get a complex if you prefer a hypercohomology point of view). Applying this for the object which represents $K(\mathbb Z(n),2n)$ (as explained in there), if $S$ is smooth over a field, one of the deepest and less trivial result of Voevodsky is that we obtain an simplicial sheaf which satisfies Nisnevich descent and is $\mathbb A^1$-homotopy invariant, so that it represents motivic cohomology both in $\mathrm H(S)$ and in $\mathrm{Ho}(sE(S))$.

If I may suggest an exercise: apply Morel and Voevodsky's construction to describe usual algebraic topology: instead of the Nisnevich site of smooth $S$-scheme, consider the site of smooth analytic manifold on C (with the usual topology) (and replace the affine line by the disk $\mathbb D^1$). Then Morel and Voevodsky theory gives a category which is canonically equivalent to the usual homotopy theory of topological spaces (this is due to the fact that any smooth complex manifold is locally constractible, so that after trivializing $\mathbb D^1$, only locally constant invariants remain). Then, for instance, Poincaré lemma says that the de Rham complex is $\mathbb D^1$-homotopy invariant. In this precise sense, this shows that complex de Rham cohomology is very well defined on any homotopy type.


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