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

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