论文标题
Hermitian K理论的稳定$ \ infty $ - 类别II：COBORDISM类别和添加性
Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity
论文作者
论文摘要
我们在Poincaré$ \ infty $ - 类别的环境中定义了Grothendieck-Witt Spectra，并表明它们适合具有K-和L Wheoretic部分的扩展。作为后果，我们推断了Verdier商的本地化序列，以及Karoubi的基本和周期性定理的概括，其中2不需要可逆。我们的设置允许对这种代数示例进行统一处理以及同质理论概括：例如，周期性定理适用于复杂的$ \ Mathrm {e} _1 $ - 环，我们表明，Grethendieck-witt的参数性光谱理论恢复了Weiss and Williams和Williams'La la la theory la theory la theore'theory。 我们的Grothendieck-Witt Spectra是通过Hermitian Q-sonstruction的版本来定义的，我们方法的新颖特征是将后者解释为COBORDISM类别。这种观点还使我们能够给出Blumberg，Gepner和Tabuada定理的Hermitian版本以及简洁的证据，并提供了上述LA-Spectra的Coobordism理论描述。
We define Grothendieck-Witt spectra in the setting of Poincaré $\infty$-categories and show that they fit into an extension with a K- 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 holds for complex oriented $\mathrm{E}_1$-rings, and we show that the Grothendieck-Witt theory of parametrised spectra recovers Weiss and Williams' LA-theory. Our Grothendieck-Witt spectra are defined via a version of the hermitian Q-construction, and a novel feature of our approach is to interpret the latter as a cobordism category. This perspective also allows us to give a hermitian version -- along with a concise proof -- of the theorem of Blumberg, Gepner and Tabuada, and provides a cobordism theoretic description of the aforementioned LA-spectra.
