Infinity categories with duality and hermitian multiplicative infinite loop space machines

TL;DR

This paper develops a framework connecting preadditive infinity categories with duality to hermitian K-theory spectra, enabling the construction of E-infinity ring spectra and providing new examples and classifications of dualities.

Contribution

It introduces a method to produce hermitian K-theory spectra from preadditive infinity categories with duality and classifies dualities in these contexts.

Findings

Hermitian K-theory spectra arise from preadditive infinity categories with duality.

Any preadditive symmetric monoidal infinity category with duals admits a canonical duality.

Applications include finitely generated projective modules over E-infinity ring spectra.

Abstract

We show that any preadditive infinity category with duality gives rise to a direct sum hermitian K-theory spectrum. This assignment is lax symmetric monoidal, thereby producing E-infinity ring spectra from preadditive symmetric monoidal infinity categories with duality. To have examples of preadditive symmetric monoidal infinity categories with duality we show that any preadditive symmetric monoidal infinity category, in which every object admits a dual, carries a canonical duality. Moreover we classify and twist the dualities in various ways and apply our definitions for example to finitely generated projective modules over E-infinity ring spectra.