Partial parametrized presentability and the universal property of equivariant spectra

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.

arXiv commentMinor revisions in response to referee report. 63 pages

References in corpus (10)

• Grothendieck-Neeman duality and the Wirthmüller isomorphism
• Parametrized higher category theory
• On equivariant homotopy theory for model categories
• Mackey 2-functors and Mackey 2-motives
• Symmetric spectra model global homotopy theory of finite groups
• Left-exact Localizations of $\infty$-Topoi I: Higher Sheaves
• Equivariant infinite loop space theory, the space level story
• Global homotopy theory via partially lax limits
• G-Global Homotopy Theory and Algebraic K-Theory
• Parametrised Presentability over Orbital Categories