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The objects of interest in motivic homotopy theory are "spaces"-which are simplicial sheaves of sets on the big Nisnevich site $Sm/k$ of smooth schemes of finite type over a field $k$. I understand that we need to enlarge the category of smooth schemes over $k$ to spaces, since $Sm/k$ is not closed under limits and colimits.

Why does one use Nisnevich topology? If one considers the Zariski site over $Sm/k$, is the category of simplicial sheaves over it not complete/co-complete? Can someone please give an example demonstrating this? Is this the only reason for using Nisnevich topology (apart from the fact that it retains useful properties of both Zariski and etale topologies)?

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  • $\begingroup$ The category of sheaves over any (essentially small) site is complete and cocomplete. This is hardly a special property of the Nisnevich site. On the other hand, a complete and cocomplete category is not enough to do homotopy theory either. $\endgroup$ – Zhen Lin Jan 24 '14 at 16:01
  • $\begingroup$ @Zhen: I agree with you. Of course, for a well-working homotopy theory, one needs a model structure. One of them used in motivic homotopy theory is the so-called injective local model structure. What is, then, the specialty of Nisnevich topology? Is it the reason (and if yes, how) for making this model structure work? $\endgroup$ – Carl Jan 24 '14 at 16:12
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    $\begingroup$ A priori geometric reasons: it retains useful properties of both Zariski (Brown-Gersten property to check that a presheaf is a sheaf/that a simplicial presheaf is fibrant, cohomological dimension bounded by Krull dimension on noetherian schemes) and \'etale (local structure of smooth pairs). A posteriori reasons: classical invariants like algebraic K-theory and higher Chow groups become representable, while the invariants you get with the parallel \'etale theory are different. $\endgroup$ – Simon Pepin Lehalleur Jan 24 '14 at 17:31
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    $\begingroup$ You need a topology not weaker than the Nisnevich one in order to prove the homotopy purity theorem of Morel-Voevodsky. On the other hand, it's reasonable to have a topology such that the spectra of arbitrary fields are points. $\endgroup$ – Mikhail Bondarko Jan 24 '14 at 19:19
  • $\begingroup$ @SimonPepinLehalleur: Thanks for your comment. Would you please elaborate a bit on "local structure of smooth pairs"? $\endgroup$ – Carl Jan 25 '14 at 11:37
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Here are some comments about the use of topologies in motivic homotopy theory. This is based on the discussion in Morel-Voevodsky's "A^1-homotopy theory of schemes" p.94-95 (MV below), I only add some background and references. I also comment on the differences between the development of the unstable and stable theories. I am not an expert, so please comment/edit out innacurate statements.

First some model category generalities. Various model structures on categories of simplicial (pre)sheaves on any Grothendieck site $(\mathcal{C},\tau)$ can be constructed. For a survey see:

http://nlab.mathforge.org/nlab/show/model+structure+on+simplicial+presheaves

and the references therein. Many of those are Quillen equivalent and can be thought as different presentations of the $(\infty,1)$-category of $(\infty,1)$ $\tau$-sheaves on $\mathcal{C}$. As usual in model category theory, they are suited for deriving different functors (i.e. some natural functors will be Quillen with respect to some structures but not others).

The resulting model categories are all left proper, simplicial and combinatorial, so by a theorem of Smith (see http://nlab.mathforge.org/nlab/show/Bousfield+localization+of+model+categories#Existence )) left Bousfield localizations at a set of maps exist. In particular if $\mathcal{C}$ is a small category of schemes over a fixed $S$ which is stable by $S$-fiber products and contains $\mathbb{A}^1_S$, one can localize at the set of maps $X\times_S \mathbb{A}^1_S\rightarrow X$.

The Morel-Voevodsky category $\mathcal{H}(S)$ can be obtained by this procedure with $\mathcal{C}=Sm/S$, $\tau=Nis$ and using as starting point the so-called injective model structure on simplicial sheaves. It should be clear from the above that many variants are possible, some giving alternative models for $\mathcal{H}(S)$ (using projective model structures, simplicial presheaves instead of sheaves, or even more exotic choices like cubical presheaves, etc.) and some giving different categories (using another topology: Zariski, étale, using the category of all schemes instead of smooth ones, etc.).

The passage to the stable theory via model categories of spectra is also formal (in the sense that it can be done in great generality and with variants) but rather subtle. See e.g. Riou "Catégorie homotopique stable d'un site suspendu avec intevalle" or Ayoub's thesis.

The question now becomes: in which respects are $\mathcal{H}(S)$ and $\mathcal{SH}(S)$ nicer than the alternatives ? And how much does this depend on the choice of the Nisnevich topology ?

Here are some possible answers:

1) Characterisation of $\tau$-local simplicial (pre)sheaves. For a general site (and in particular for the étale site over a general scheme), this is a complicated condition which can be expressed only in terms of a descent condition for hypercovers (see Dugger-Hollander-Isaksen, "Hypercovers and simplicial presheaves"). For the Zariski and the Nisnevich site, one can show a "Brown-Gersten" property: $\tau$-locality can be rephrased as some squares of simplicial sets associated to $\tau$-distinguished squares being homotopy cartesian. See MV proposition 1.16 for the case of the Nisnevich topology.

Aside: this argument has been abstracted by Voevodsky in "Homotopy theory of simplicial sheaves in completely decomposable topologies" and used to compare $\mathcal{H}(k)$ for $k$ a field admitting resolution of singularities with a similar category defined with $\mathcal{C}=Sch/k$ and $\tau=cdh$. In the setting of triangulated categories of mixed motives, this argument can be pushed in various directions to exploit various forms of resolution of singularities, see Cisinski-Deglise, "Triangulated categories of mixed motives", section 3.3.

The Brown-Gersten property in turn plays an important role in MV. For instance, it implies that the property of being $\tau$-local is stable by filtered colimits. It is also used in MV to construct an explicit A^1-localisation functor and to study the functoriality of $\mathcal{H}(S)$. On the other hand I am not sure if the use of the Brown-Gersten property for all this is unavoidable: at least in the context of $\mathcal{SH}(S)$, there are arguments which do not use this (see Riou's paper mentioned above or Ayoub's thesis) and which consequently give some results for the étale topology. On the other hand, in following works of Morel on $\mathcal{H}(k)$, there are really substantial applications of the Brown-Gersten condition, see "A^1-algebraic topology over a field" Chap. 8 and App. A.

2) Homotopy purity (or localisation). The proof of this major theorem in MV requires a topology at least as fine as the Nisnevich topology and to work with $\mathcal{C}=Sm$. The idea is that to reduce the theorem to the case of the closed immersion $Z\rightarrow A^n_Z$ (where one can write explicit $A^1$-homotopies), one uses the local structure of smooth pairs in the étale topology (cf EGAIV 17.12.2) and the fact that étale morphisms to Henselian local schemes (i.e. points of the Nisnevich site) which have a section on the closed point have a section.

3) Compactness properties. The cohomological dimension of the small Nisnevich site on a noetherian scheme is bounded by the Krull dimension, see MV 1.8 and Thomason-Trobaugh E.6.c. This is very different from the case of the étale topology. This has important consequences for the stable theory. It implies in particular that $\mathcal{SH}(S)$ is compactly generated. See e.g. the last paragraphs of Ayoub's thesis. I do not know if there are analoguous statements to be made in the unstable case.

4) Nisnevich descent for motivic cohomology and algebraic K-theory. The descent properties of algebraic K-theory have been studied long before motivic homotopy theory. The definitive, pre-$A^1$-homotopy result (Nisnevich descent for algebraic K-theory of regular schemes) is in Thomason-Trobaugh, and this is used as an input in the proof of representability of algebraic K-theory in MV p.139. On the other hand, algebraic K-theory with integral coefficients does not satisfy étale descent and hence cannot be represented in the étale $A^1$-homotopy category.

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  • $\begingroup$ Regarding your point about compactness properties, is it true that $SH(S)$ is compactly generated, if one considers étale site over $S$. If not true generally , is it known under what conditions over $S$ it happens (one condition maybe that Postnikov tower converges) $\endgroup$ – iron feliks Sep 26 '18 at 20:17
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    $\begingroup$ @iron felik In MV $A^1$-homotopy category, Postnikov tower always converges for $S$ a noetherian finite dimensional scheme. This is done in [MV, 2.1.37]; see also Lurie [DAG XI, Theorem 2.24]. $\endgroup$ – Lao-tzu Nov 28 '18 at 20:19

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