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 $${\text {K}}$$ 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 $${\text {L}}$$ 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 $${\text {L}}$$ L - and algebraic $${\text {K}}$$ K -groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.

Witt groups of Spinor varieties

“…For 𝑛 even there are two classes of even shifted Young diagrams: those whose first row is full and those whose last column is empty. Note that this fact can be viewed as a diagrammatic counterpart to the recursive description given in Formula (7).…”

“…There are also recent developments on the Hermitian $$K$$‐theory when 2 is not invertible in the base, cf. [7] and [31]. Investigating the Hermitian $$K$$‐theory of schemes in these frameworks seems to be an interesting project.…”

The family signature theorem

“…Ranicki's work in this direction was visionary, but his specific technical implementation is not ideal for our purposes (see remark 4.9). Instead, we will use the recent framework of Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle [10][11][12].…”

“…In subsections 4.5-4.8 we will make extensive use of the definitions and constructions of [10][11][12]. We will give references for all results that we use, but the reader will need to be familiar with the general set-up of those papers, which we will not review.…”

“…This will require significantly more external machinery than the discussion so far. In particular it relies extensively on the relation between L-theory and Grothendieck-Witt theory developed in [10][11][12], and we must now assume more familiarity with these papers.…”