Infinity categories with duality and hermitian multiplicative infinite loop space machines
Hadrian Heine, Alejo Lopez-Avila, Markus Spitzweck
TL;DR
The paper builds a cohesive theory of dualities in ∞-categories and their impact on K-theory, introducing direct sum hermitian K-theory for preadditive symmetric monoidal ∞-categories with duality and showing it yields $E_\infty$-ring spectra; it refines to real K-theory as a genuine $C_2$-spectrum (KR) and provides a canonical, canonical-section-based construction of dualities in rigid monoidal settings. Central to the work is the interpretation of dualities as $C_2$-fixed points and pro-dualities, and the development of a robust Day-convolution/Span-theoretic framework enabling lax and genuine monoidal structures. The results illuminate the landscape of dualities in spectra and module categories, enabling systematic construction of hermitian and real K-theories and revealing a rich space of dualities, including uncountably many on compact spectra. Together, these insights advance how duality interacts with monoidal ∞-categories, K-theory, and equivariant refinements, with concrete implications for modules over $E_\infty$-ring spectra and related algebraic K-theory constructions.
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.