$(\infty,\infty)$-Categorical Universal Motives

Abstract

Ever since the introduction of motivic homotopy theory, as a well-proposed approximation of Grothendieck's dream, algebraic geometers then have the chance to study schemes via a homotopy theory. However topologists also found that lifting the usual homotopy theory over a sphere spectrum to the motivic homotopy category over a motivic bigraded sphere spectrum can make breakthroughs on elementary topology problems (such as computing homotopy groups of spheres, motivic Adams spectral sequences and so on). On the other hand, topological spaces can be all regarded as Grothendieck topoi, as in Lurie's work on ultracategories. Following Scholze, Lurie we systematically consider an $(\infty,\infty)$-ultracategorical universal motivic formalism, which directly fits into Lurie's framework on ultracategories, where we construct universal ultragestalten through motivicalization. The gestalten higher categorical six-functor formalism then allows us to consider the following classes of problems of different flavors: (I) Six-functor formalism for all Grothendieck sites and Grothendieck topoi; (II) Six-functor formalism for all topological spaces. We then in this paper got the chance to apply this to many problems in $p$-adic geometry and $p$-adic functional analysis: (I) $(\infty,\infty)$-categoricalization of motivic $+$-de Rham prismatization approach to generalization of Colmez's Montréal functor; (II) $(\infty,\infty)$-categoricalization of motivic generalized Riemann-Hilbert correspondence after Bhatt-Lurie; (III) $(\infty,\infty)$-categorical universal motives of derived algebraic stacks over $\mathbb{E}_\infty$-ring objects in the derived $\infty$-category of sphere spectrum, via universal motives of large Fargues-Fontaine gestalten, in families.