Free duals and a new universal property for stable equivariant homotopy theory

TL;DR

This paper introduces a universal property for stable equivariant homotopy theory, showing how genuine G-spectra can be constructed from naive spectra by freely adjoining duals, with a focus on free constructions and product decompositions.

Contribution

It establishes a new universal property for symmetric monoidal ∞-categories with duals and finite colimits, and applies it to characterize genuine G-spectra as free extensions of naive spectra.

Findings

The left adjoint functor splits as a product of three factors with universal properties.

Genuine G-spectra are obtained from naive G-spectra by freely adjoining duals for compact objects.

The construction respects colimits and provides a new perspective on equivariant stable homotopy theory.

Abstract

We study the left adjoint $\mathbb{D}$ to the forgetful functor from the $\infty$-category of symmetric monoidal $\infty$-categories with duals and finite colimits to the $\infty$-category of symmetric monoidal $\infty$-categories with finite colimits, and related free constructions. The main result is that $\mathbb{D} \mathcal C$ always splits as the product of 3 factors, each characterized by a certain universal property. As an application, we show that, for any compact Lie group $G$, the $\infty$-category of genuine $G$-spectra is obtained from the $\infty$-category of Bredon (\emph{a.k.a} ``naive") $G$-spectra by freely adjoining duals for compact objects, while respecting colimits.