Hermitian K-theory for stable infty -categories I: Foundations

Abstract

This paper is the first in a series in which we offer a new framework for hermitian {text {K}}-theory in the realm of stable infty -categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré infty -category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraic Thom construction. For derived infty -categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on infty -categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré infty -categories, showing in particular that they form a bicomplete, closed symmetric monoidal infty -category. We also study the process of tensoring and cotensoring a Poincaré infty -category over a finite simplicial complex, a construction featuring prominently in the definition of the {text {L}}- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré infty -category using generators and relations. We extract its basic properties, relating it in particular to the 0th {text {L}}- and algebraic {text {K}}-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.