Levels of algebraicity in stable homotopy theories

TL;DR

This paper explores various notions of algebraicity in stable homotopy theory, revealing complex relationships and illustrating them with examples from chromatic homotopy theory.

Contribution

It clarifies the connections between different algebraic concepts in stable homotopy theory and provides new insights through illustrative examples.

Findings

Different notions of algebraicity are subtly related.

Implications between algebraic notions are established.

Examples from chromatic homotopy theory demonstrate these relationships.

Abstract

We study several different notions of algebraicity in use in stable homotopy theory and prove implications between them. The relationships between the different meanings of algebraic are unexpectedly subtle, and we illustrate this with several interesting examples arising from chromatic homotopy theory.