• Corpus ID: 221703712

Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity

@article{Calms2020HermitianKF,
  title={Hermitian K-theory for stable \$\infty\$-categories II: Cobordism categories and additivity},
  author={Baptiste Calm{\`e}s and Emanuele Dotto and Yonatan Harpaz and Fabian Hebestreit and Markus Land and Kristian Jonsson Moi and Denis Nardin and Thomas Nickelsen Nikolaus and Wolfgang Steimle},
  journal={arXiv: K-Theory and Homology},
  year={2020}
}
We define Grothendieck-Witt spectra in the setting of Poincare $\infty$-categories and show that they fit into an extension with a L- and an L-theoretic part. As consequences we deduce localisation sequences for Verdier quotients, and generalisations of Karoubi's fundamental and periodicity theorems for rings in which 2 need not be invertible. Our set-up allows for the uniform treatment of such algebraic examples alongside homotopy-theoretic generalisations: For example, the periodicity theorem… 
On Atiyah-Segal completion for T-equivariant Hermitian K-theory
We show how derived completion can be used to prove an analogue of Atiyah-Segal completion for the T -equivariant Hermitian K-theory of a scheme X with a trivial T -action, containing 12 and
Generalized cohomology theories for algebraic stacks
We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck’s six operations. Objects in this
Hermitian K-theory via oriented Gorenstein algebras
We show that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along finite Gorenstein morphisms with trivialized dualizing sheaf. As an application, we
On k-invariants for $(\infty, n)$-categories
Every $(\infty, n)$-category can be approximated by its tower of homotopy $(m, n)$-categories. In this paper, we prove that the successive stages of this tower are classified by k-invariants,
On automorphisms of high-dimensional solid tori
We study the infinite generation in the homotopy groups of the group of diffeomorphisms of $S^1 \times D^{2n-1}$, for $2n \geq 6$, in a range of degrees up to $n-2$. Our analysis relies on
On the rational motivic homotopy category
We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of
On the homotopy type of L‐spectra of the integers
We show that quadratic and symmetric L ‐theory of the integers are related by Anderson duality and that both spectra split integrally into the L ‐theory of the real numbers and a generalised

References

SHOWING 1-10 OF 29 REFERENCES
The Picard group in equivariant homotopy theory via stable module categories
We develop a mechanism of "isotropy separation for compact objects" that explicitly describes an invertible $G$-spectrum through its collection of geometric fixed points and gluing data located in
The transfer and stable homotopy theory
  • D. S. Kahn, S. Priddy
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1978
The purpose of this paper is to give a proof of the following splitting theorem in stable homotopy theory. We assume all spaces are localized at a fixed prime p. Let k be the symmetric group on {1,
Hermitian K -theory of exact categories
We study the theory of higher Grothendieck-Witt groups, alias algebraic hermitian K-theory, of symmetric bilinear forms in exact categories, and prove additivity, cofinality, devissage and
Excision in cyclic homology and in rational algebraic $K$-theory
(for a precise definition, see ?1 below). By replacing everywhere K*( ) by K*( ) ? Q, one obtains the corresponding notion in rational algebraic K-theory. The above definition has an obvious
K-Theory of Forms.
The homotopy type of the cobordism category
The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main
Invariance de la K-théorie par équivalences dérivées
The aim of these notes is to prove that any right exact functor between reasonable Waldhausen categories, that induces an equivalence at the level of homotopy categories, gives rise to a homotopy
...
...