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Motivic functors. (English) Zbl 1042.55006

The authors give a model for the Morel-Voevodsky motivic stable category which is loosely based on the Lydakis approach to the construction of the ordinary stable category.
The basic building blocks of the theory are the motivic functors, which are enriched functors from the finitely presented objects in the category of motivic spaces to the full category of motivic spaces. Motivic spaces are pointed simplicial presheaves on the smooth Nisnevich site; maps between them are motivic weak equivalences if they are local weak equivalences for the Nisnevich topology and do not see the affine line in a suitable sense. Motivic functors restrict to spectrum and/or symmetric spectrum objects, and as such define objects in the motivic stable category. One of the main features of the category of motivic functors is that it has an easily defined symmetric monoidal smash product.
The main result of the paper asserts that, for a suitable closed model structure on the category of motivic functors, this category is Quillen equivalent to the symmetric spectrum model for the Morel-Voevodsky stable category.

MSC:

55P42 Stable homotopy theory, spectra
14F42 Motivic cohomology; motivic homotopy theory
19E20 Relations of \(K\)-theory with cohomology theories
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