# Monochromatic homotopy theory is asymptotically algebraic

## TL;DR

This paper extends previous ultraproduct-based results to show that the symmetric monoidal $$-categories of $K_{p}(n)$-local spectra are asymptotically algebraic in the prime $p$, using advanced $$-categorical tools.

## Contribution

It proves that $K_{p}(n)$-local spectra categories are asymptotically algebraic, complementing earlier results for $E_{n,p}$-local spectra, with compatible equivalences.

## Key findings

- $K_{p}(n)$-local spectra categories are asymptotically algebraic.
- The results are compatible with $E_{n,p}$-local spectra equivalences.
- Introduces $$-categorical tools for non-compact units.

## Abstract

In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $\infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $\infty$-categorical tools suitable for working with compactly generated symmetric monoidal $\infty$-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the $E_{n,p}$-local $\infty$-categories.