Partial parametrized presentability and the universal property of equivariant spectra

TL;DR

This paper develops a theory of partial presentability in parametrized higher categories, constructs free examples, and applies these to show that the global category of genuine equivariant spectra is the free equivariantly presentable and stable category.

Contribution

It introduces partial presentability in parametrized higher categories and demonstrates its application to equivariant homotopy theory, establishing the universal property of genuine equivariant spectra.

Findings

Global category of genuine equivariant spectra is the free equivariantly presentable and stable category.

Constructs free partially presentable categories in various contexts.

Shows that the global category of genuine equivariant spectra has a universal property.

Abstract

We introduce a notion of partial presentability in parametrized higher category theory and investigate its interaction with the concepts of parametrized semiadditivity and stability from arXiv:2301.08240. In particular, we construct the free partially presentable $T$-categories in the unstable, semiadditive, and stable contexts and explain how to exhibit them as full subcategories of their fully presentable analogues. Specializing our results to the setting of (global) equivariant homotopy theory, we obtain a notion of equivariant presentability for the global categories of arXiv:2301.08240, and we show that the global category of genuine equivariant spectra is the free global category that is both equivariantly presentable and equivariantly stable. As a consequence, we deduce the analogous result about the $G$-category of genuine $G$-spectra for any finite group $G$, previously formulated by Nardin.