$\infty$-Categorical Functional Analysis and $p$-adic Motives

TL;DR

This paper explores the connection between $ abla$-categorical functional analysis and $p$-adic motives, aiming to bridge homotopical methods with arithmetic applications in a novel theoretical framework.

Contribution

It introduces a new perspective linking $ abla$-categorical analysis with $p$-adic motives, expanding the foundational understanding of $ abla$-categories in arithmetic geometry.

Findings

Establishes a conceptual framework connecting $ abla$-categories and $p$-adic motives.

Builds on recent advances in $ abla$-categorical homotopical methods.

Proposes potential applications in arithmetic and number theory.

Abstract

We discuss the deep relationship between $\infty$-categorical functional analysis and the anticipated theory of $p$-adic motives. The motivation fundamentally comes from applications essentially in arithmetics from very broad perspectives. The $\infty$-categorical homotopical aspects we considered here come mainly from Bambozzi-Ben-Bassat-Kremnizer, Ben-Bassat-Mukherjee, Bambozzi-Kremnizer, Clausen-Scholze and Kelly-Kremnizer-Mukherjee, based on deep robust foundation of $\infty$-categorical solids and $\infty$-categorical ind-Banach or bornological modules over general Banach rings.