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I am currently trying to study $\mathbb{A}^1$-homotopy theory and I have a question about the construction of the unstable motivic category.

Here is roughly the construction I try to understand :

1) Fix a noetherian scheme of finite Krull dimension $S$, and denote the category of smooth schemes of finite type over it by $\text{Sm}/S$. It is (essentially) small due to the finiteness condition. Endow it with the Nisnevich topology. OK.

2) The category of simplicial presheaves $[\text{Sm}/S^{\text{op}}, \text{sSet}]$, denoted by $M_S$, has (at least) 3 local model structures, the injective (Jardine), the projective (Dugger, Hollander, Isaksen) and the flasque (Isaksen). The weak equivalences in these structures are the local weak equivalences, which can be characterized either by being the maps inducing isomorphisms on all sheaves of homotopy groups (Jardine), or the maps inducing isomorphisms on all stalks. These models can be seen as a left Bousfield localization of the global model structures. Similar models hold by restricting to simplicial sheaves instead of presheaves, and they are Quillen equivalent by sheafification-embedding. OK.

3) There still is a localization to be done, and this is the one I don't quite understand. It is the $\mathbb{A}^1$-localization, or the localization with respect to the interval. Here is what I understand :

In "$\mathbb{A}^1$-homotopy theory of schemes" of Morel and Voevodsky : They do a more general construction for any site with interval. Everything is done "by hand", Theorem 2.2.5 is the left Bousfield localization, Theorem 2.3.2 is the localization of the simplicial sheaves "with respect to the interval", and Definition 3.2.1 is the case of interest, the site $\text{Sm}/S$ with the interval $\mathbb{A}^1$. Their model structure seems to be a left Bousfield localization of the category of simplicial sheaves at the unique map $\mathbb{A}^1 \to \ast$. Magically, all the projections $\mathbb{A}^1 \times F \to F \in M_S$ are weak equivalences ? So the localization can be formally done by apllying a left Bousfield localization ?

Moreover, this is done on simplicial sheaves, does a similar result hold for simplicial presheaves ? I would be very happy to hear that in the left Bousfield localization of some local model structure on $[\text{Sm}/S^{op},sSet]_{\text{Nis}}$ at the unique map $\mathbb{A}^1 \to \ast$, all the maps $\mathbb{A}^1 \times F \to F$ of simplicial presheaves are weak equivalences. Moreover, is this the property we want in the unstable category of motivic spaces ? We could of course try to do a left Bousfield localization at all maps $\mathbb{A}^1 \times F \to F$, but since it is not a set, this does not necessarily exists, a priori.

Thanks. Feel free to redirect me to any reference.

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up vote 4 down vote accepted

Markus Severitt wrote a wonderfully readable introduction to the unstable motivic category, and it seems to me he answers all of your questions. In particular, look at section 6.2 of his paper.

http://www.math.uni-bielefeld.de/~mseverit/mseverittse.pdf

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Thank you for this. It actually helped me understand many points. However, there still is one question that I have. Are all the projections $\mathbb{A}^1 \times F \to F$ weak equivalences (for any simplicial presheaf $F$ and not only a representable) ? –  redfiloux May 27 '12 at 17:18
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