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

• Baptiste Calmès ^[1] ; Emanuele Dotto ^[2] ; Yonatan Harpaz ^[3] ; Fabian Hebestreit ^[6] ; Markus Land ^[7] ; Kristian Moi ^[8] ; Denis Nardin ^[4] ; Thomas Nikolaus ^[9] ; Wolfgang Steimle ^[5]
  1. [1] Artois University

     Artois University

     Arrondissement d’Arras, Francia

     
  2. [2] University of Warwick

     University of Warwick

     Reino Unido

     
  3. [3] Paris 13 University

     Paris 13 University

     Arrondissement de Saint-Denis, Francia

     
  4. [4] University of Regensburg

     University of Regensburg

     Kreisfreie Stadt Regensburg, Alemania

     
  5. [5] University of Augsburg

     University of Augsburg

     Kreisfreie Stadt Augsburg, Alemania

     
  6. [6] RFWU Bonn, Germany
  7. [7] LMU München, Germany
  8. [8] Institutionen för matematik, Sweden
  9. [9] WWU Münster, Germany

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 1, 2023
• Idioma: inglés
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  • This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable ∞-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 ∞-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 ∞-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 ∞-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 ∞-categories, showing in particular that they form a bicomplete, closed symmetric monoidal ∞-category. We also study the process of tensoring and cotensoring a Poincaré ∞-category over a finite simplicial complex, a construction featuring prominently in the definition of the 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 ∞-category using generators and relations. We extract its basic properties, relating it in particular to the 0th L- and algebraic K-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.