Rational homotopy theory is a simplification of homotopy theory. Formally, the class of weak equivalences is increased by the collection of maps which induce an isomorphism on singular (co)homology with rational coefficients. Work of Quillen, Sullivan and Goerss shows that a nice subcategory of the homotopy category of rational spaces is completely algebraic; in other words, there is a fully faithful functor from such subcategory to a homotopy category of algebraic nature. In this talk, I will describe the analogous problem for the category of motivic spaces. In this setting, we have several candidates to play the role of singular homology, named A¹-homology, Suslin homology and motivic cohomology. Concretely, I will present a model for rationalization using A1-homology.

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