Chromatic homotopy theory is asymptotically algebraic

TL;DR

This paper introduces categorical ultraproducts to analyze the asymptotic behavior of chromatic homotopy theory, demonstrating that at a fixed height, it becomes equivalent to algebraic categories, thus providing an asymptotic algebraic perspective.

Contribution

It develops a new framework of categorical ultraproducts inspired by the Ax--Kochen theorem to connect chromatic homotopy theory with algebraic categories asymptotically.

Findings

Ultraproducts of $E(n,p)$-local categories are equivalent to algebraic categories.

Chromatic homotopy theory at fixed height is asymptotically algebraic.

Provides an asymptotic solution to the approximation problem in chromatic homotopy theory.

Abstract

Inspired by the Ax--Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the $E(n,p)$-local categories over any non-prinicipal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically algebraic.