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

Abstract

Abstract This paper is the first in a series in which we offer a new framework for hermitian $${\text {K}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>K</mml:mtext></mml:math> -theory in the realm of stable $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> -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 $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> -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 $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> -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 $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> -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 $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> -categories, showing in particular that they form a bicomplete, closed symmetric monoidal $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> -category. We also study the process of tensoring and cotensoring a Poincaré $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> -category over a finite simplicial complex, a construction featuring prominently in the definition of the $${\text {L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>L</mml:mtext></mml:math> - and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> -category using generators and relations. We extract its basic properties, relating it in particular to the 0th $${\text {L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>L</mml:mtext></mml:math> - and algebraic $${\text {K}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>K</mml:mtext></mml:math> -groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.

Topics & Concepts