All Questions

For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it: (a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here ...

There is a really interesting comment in this question that I was unable to find a reference... Under the "Tate conjectures, then every motive belongs to the tensor category generated by motives of ...

I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one. For the algebraic cobordism theory $...

A slight variant $\tilde Mot_{num}(k,\mathbb{Q})$ of the category of pure motives $Mot_{num}(k,\mathbb{Q})$ is a Tannakian category equivalent to a category of representations of some algebraic group $...

First I would like to know if this has been worked out, and if the answer is affirmative I would like to know some references.

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...