Modular Model Categories

Abstract

To any model category $\mathcal{M}$, we associate a modular model category, a functor of points $\mathcal{M}[-]:$ Cat $\to$ 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.

31 pages

References in corpus (1)

• Motivic classifying $\infty$-topoi and spectral stacks