My question is not very precise, I apologize in advance. Essentially I am wondering in the context of "homotopy theory for schemes, in the broad sense, be it in motivic homotopy theory or $\mathbb{A}^1$-homotopy theory, why do we use simplicial (pre)sheaves over the category of smooth schemes.
I think I understand the "very big picture", and also some of the minute details of the theory. But what I'm failing to understand is in between.
Essentially the category of schemes is badly behaved with respect to colimits, for instance one may not take a quotient as freely as in the category of topological spaces or CW-complexes.
If I try to think about how I would go about curcumventing that problem, the natural thing that comes to mind is to consider simplicial schemes i.e you glues schemes along algebraic maps, but since the latter is no longer a scheme in general you keep the pieces separated and just keep track of the "glueing recipe". Sorry this is a little bit of hand waving, but this is also the kind of answer I'm looking for.
But this is not what people do. Instead the consider simplicial presheaves on the category of say, smooth finite type schemes over a base field. I understand that this has very nice categorical properties, and one can construct model catgeories on these which give us good homotopy theory but i fail to understand why these objects are the right one, and why one should consider these in the first place.
The definition of a motivic space feels a bit puzzling to me as it feels that there are way too many motivic spaces than I would like.
Of course we have schemes via the Yoneda embedding, and these are obvisouly fine. But what sense to make of "constant" simplicial sets? By that I mean take $X$ any simplicial set, and define $X(\bullet)$ to be the presheaf $X(T):=X$ for any scheme $T$. I fail to make sense of these objects, and why they should be here.
One reasons usually people give is the bigrading. The bigrading is brought out by the existence of two obvious circles. The Tate circle, and the simplicial circle. I do realize that this is important, but I still don't get a good idea of what this simplicial circle is geometrically, and why it should be there. I mean I do understand what is the simplicial circle, from a simplicial set point of view, it's just the regular old circle.
The only, rather vague, idea I have to make sense of this is that those additionnal simplicial sets we add up, are where we construct homotopies. If $X$ is an algebraic variety, then $X\times I=X\times [0,1]$ certainly isn't, but if one wants to do homotopy theory with $X$ then certainly one needs that object at some point. So I'm gessing this is why we have this mixed thoery where we can take a product of a scheme and a simplicial set which is definitely not algebraic.
However, in motivic homotopy theory, the part playing the role of $I$ is $\mathbb{A}^1$ so I'm not even sure that my intuition is correct about that. After all, we do render the theory $\mathbb{A}^1$-local when we construct models on the category of motivic spaces/spectra.
Should I just accept that these objects are the right ones, essentially because in the end we have what we want? Namely the ability to form motivic spectra and to build cohomology theories out of those etc....
Sorry if that question is not very precise, I just would like to "get" why we consider this is the "right category" to consider. I also completely left out the Nisnevisch topology out of the picture.