To any model category $\mathcal{ M}$, we associate a modular model category, a functor of points $\mathcal{ M}[-]:$ Cat $\rightarrow$ Cat, that associates to any small category $\mathcal{ C}$ a functor category $\mathcal{ M}[\mathcal{ C}] = \text{ Fun}_{ fes}(\mathcal{ C}, \mathcal{ M})$ of full and essentially surjective functors from $\mathcal{ C}$ to $\mathcal{ M}$, providing parametrizations of a same model category $\mathcal{ M}$ by different small categories. We are in particular interested in using schemes as parameters. We consider $\mathbb{ Z}$Sm$/k$ the category of linear combinations of smooth separated schemes of finite type over Spec($k$), $k$ a field, referred to as $\mathbb{ Z}$-schemes, and let $\mathcal{ C} = Sh(\mathbb{ Z} \text{ Sm}/k, \text{ Nis})$. We contrast this with using the $\mathbb{ A}^1$-homotopy category of $\mathbb{ Z}$-schemes as a parametrizing category.
Categories: math.AG