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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.

Let G be a finite group acting on a small category I. We study functors X : I → C equipped with families of compatible natural transformations that give a kind of generalized G-action on X. Such objects are called G-diagrams. When C is a sufficiently nice model category we define a model structure on the category of G-diagrams in C . There are natural G-actions on Bousfield-Kan style homotopy limits and colimits of G-diagrams. We prove that weak equivalences between point-wise (co)fibrant G-diagrams induce weak G-equivalences on homotopy (co)limits. A case of particular interest is when the indexing category is a cube. We use homotopy limits and colimits over such diagrams to produce loop and suspension spaces with respect to permutation representations of G. We go on to develop a theory of enriched equivariant homotopy functors and give an equivariant "linearity" condition in terms of cubical G-diagrams. In the case of G-topological spaces we prove that this condition is equivalent to Blumberg's notion of G-linearity. In particular we show that the Wirthmüller isomorphism theorem is a direct consequence of the equivariant linearity of the identity functor on G-spectra.

We define Grothendieck-Witt spectra in the setting of Poincaré ∞-categories, show that they fit into an extension with a K-and an L-theoretic part and deduce localisation sequences for Verdier quotients. As special cases we obtain generalisations of Karoubi's fundamental and periodicity theorems for rings in which 2 need not be invertible.A novel feature of our approach is the systematic use of ideas from cobordism theory by interpreting the hermitian Q-construction as an algebraic cobordism category. We also use this to give a new description of the LA-spectra of Weiss and Williams. CONTENTS IntroductionRecollection 1 Poincaré-Verdier sequences and additive functors 1.1 Poincaré-Verdier sequences 1.2 Split Poincaré-Verdier sequences and Poincaré recollements 1.3 Poincaré-Karoubi sequences 1.4 Examples of Poincaré-Verdier sequences 1.5 Additive and localising functors 2 The hermitian Q-construction and algebraic cobordism categories 2.1 The hermitian Q-construction 2.2 The cobordism category of a Poincaré ∞-category 2.3 Algebraic surgery 2.4 The additivity theorem 2.5 Fibrations between cobordism categories 2.6 Additivity in K-Theory 3 Structure theory for additive functors 3.1 Cobordisms of Poincaré functors 3.2 Isotropic decompositions of Poincaré ∞-categories 3.3 The group-completion of an additive functor 3.4 The spectrification of an additive functor 3.5 Bordism invariant functors 3.6 The bordification of an additive functor 3.7 The genuine hyperbolisation of an additive functor 4 Grothendieck-Witt theory 4.1 The Grothendieck-Witt space 4.2 The Grothendieck-Witt spectrum 4.3 The Bott-Genauer sequence and Karoubi's fundamental theorem 4.4 L-theory and the fundamental fibre square 4.5 The real algebraic K-theory spectrum and Karoubi periodicity 4.6 LA-theory after Weiss and Williams Date: September 16, 2020. 144 B.2 Schlichting's Grothendieck-Witt-spectrum of a ring with 2 invertible 146 References 150which we term the fundamental fibre square. Now, in [HM] Hesselholt and Madsen promoted the Grothendieck-Witt spectrum GW s cl ( , ) into the genuine fixed points of what they termed the real algebraic K-theory KR s cl ( , ), a genuine C 2 -spectrum. We similarly produce a functor KR ∶ Cat p ∞ ⟶ Sp gC 2 using the language of spectral Mackey functors, with the property that the isotropy separation square of KR(C, Ϙ) is precisely the fundamental fibre square above, so that in particular KR(C, Ϙ) gC 2 ≃ GW(C, Ϙ) and KR(C, Ϙ) C 2 ≃ L(C, Ϙ); here (−) gC 2 and (−) C 2 ∶ S gC 2 → S denote the genuine and geometric fixed points functors, respectively. Combined with the comparison results of [HS20] this affirms the conjecture of Hesselholt and Madsen, that the geometric fixed points of the real algebraic K-theory spectrum of a discrete ring are a version of Ranicki's L-theory.As the ultimate expression of periodicity, we then enhance our extension of Karoubi's periodicity to the following statement in the language of genuine homotopy theory: Theorem D. The boundary map of the metabolic Poincaré-Verdier sequence provides a...

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring to the homotopy C 2 -orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of ℤ, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension is an equivalence in degrees ≥ +3. As an important tool, we establish the hermitian analogue of Quillen's localisation-dévissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi. CONTENTS 10 1.1 L-theoretic preliminaries 10 1.2 Surgery for -quadratic structures 13 1.3 Surgery for -symmetric structures 24 2 L-theory of Dedekind rings 28 2.1 The localisation-dévissage sequence 28 2.2 Symmetric and quadratic L-groups of Dedekind rings 34 3 Grothendieck-Witt groups of Dedekind rings 43 3.1 The homotopy limit problem 43 3.2 Grothendieck-Witt groups of the integers 47 References 54

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 0-th Grothendieck-Witt group of a Poincaré ∞-category using generators and relations. We extract its basic properties, relating it in particular to the 0-th 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. CONTENTSCALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE 5 Monoidal structures and multiplicativity 78 5.1 Tensor products of hermitian ∞-categories 78 5.2 Construction of the symmetric monoidal structure 82 5.3 Day convolution of hermitian structures 88 5.4 Examples 95 6 The category of Poincaré categories 99 6.1 Limits and colimits 99 6.2 Internal functor categories 103 6.3 Cotensoring of hermitian categories 108 6.4 Tensoring of hermitian categories 113 6.5 Finite tensors and cotensors 116 6.6 Finite complexes and Verdier duality 121 7 Hyperbolic and metabolic Poincaré categories 123 7.1 Preliminaries: pairings and bifibrations 124 7.2 Bilinear and symmetric ∞-categories 129 7.3 The categorical Thom isomorphism 135 7.4 Genuine semi-additivity and spectral Mackey functors 146 7.5 Multiplicativity of Grothendieck-Witt and L-groups 152 References 156

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We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the p-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most p − 1. This extra functoriality allows us to extend the p-typical Witt vectors functor from commutative rings to Z 2-Tambara functors, for odd primes p. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of the real topological Hochschild homology spectrum at odd primes.