The theory of motives is an attempt to cope with the fact that there are many reasonable cohomology theories of algebraic varieties. Now, sometimes your cohomology theory does not just give you a bunch of groups/vector spaces; it gives you a full-fledged (pro-)homotopy type (though based on limited responses to this question, I think there is no formal way to functorially produce homotopy types given a cohomology theory). The question is: what should be the extension of motives to homotopy types? What are some notable works in this direction? I have heard there is something called motivic homotopy theory, is it really relevant here or it just happens to have a similar name?

More specific questions:

- First off, we need to show that usual motives are not good enough. This is probably obvious to anyone working in the field, but not everybody on this site is an expert on motives so an explicit example would be great. I think there is a rigorously constructed triangulated category of motives which is supposed but not known to be the derived category of some abelian category of motives. There is also a somewhat more pedestrian Grothendieck ring of varieties. I think the class in the Grothendieck ring does not determine the motive nor does the motive determine the class in the Grothendieck ring. Is there an example of two varieties with the same class in the Grothendieck ring which have different etale fundamental groups? An example of two varieties with the same motive in the triangulated category which have different etale fundamental groups?
- Before you can say "every Weil cohomology theory factors through motives", you have to define a Weil cohomology theory. What should the definition be for homotopy types? Has somebody written down a manageable list of axioms definining exactly what we are interested in?
- Before we can seriously talk about motivic homotopy types, we should understand on the categorical level what we expect the relevant category to be. In the case of motives, the relevant piece of category theory is Tannakian formalism, I believe (then we can say smart words like "a Weil cohomology theory is just a fiber functor blah blah"). What should the category theory look like for motivic homotopy types? A kind of non-abelian Tannakian formalism?

P.S.: yeah, I know the question is super naive, you are free to call me an idiot in the comments.

EDIT: I did not think about the coefficients when asking this. I believe Voevodsky's category makes sense with $\mathbb{Z}$-coefficients, so take that in the first question.

intothe category DM and not the other way round, so I'm not sure my comment will be useful for your question. You could also look for the notion of motivic fundamental group, I don't know if this can be made into a functor from some natural category (just as motivic cohomology can be seen as a functor from DM). $\endgroup$ – François Brunault May 6 '19 at 17:21