高等代数K理论的一个普适刻画
A universal characterization of higher algebraic K-theory
摘要：
本文在小稳定无穷范畴的背景下建立了高等代数K理论的一个普适刻画。具体地说，我们证明了连接代数K理论是通用的加法不变量，即，的普遍函子与频谱的价值，反转森田等价，保持过滤colimits，并满足瓦尔德豪森的可加性定理。同样，我们证明了非连接代数K理论是泛局部化不变量，即，普遍函子，而且满足“Mesquason-Trobaugh-Neeman”本地化定理。为了证明这些结果，我们构建和研究两个稳定的无穷大类别的“非交换动机”，一个与可加性和本地化。在这些稳定无穷范畴中，Waldhausen的S.构造对应于悬置函子，并且连接和非连接代数K理论谱变得可由球谱的非对易动机来共表示。特别是，每一个方案、栈和环谱的代数K理论都可以从这些非对易动机的范畴中恢复出来。为了与这些类别的非交换动机，我们建立了比较定理的范围内的谱范畴本地化的森田等价和范畴的小幂等完全稳定的无穷大范畴。我们还详细解释了Waldhausen K-理论的无穷范畴版本与经典定义之间的比较。作为我们理论的应用，我们得到了从高等代数K理论到拓扑霍赫希尔德同调（THH）和拓扑循环同调（TC）的自然变换的完全分类.值得注意的是，我们得到一个优雅的概念描述的分圆迹映射。
英文摘要：
In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity categories. Specifically, we prove that connective algebraic K-theory is the universal additive invariant, i.e., the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. Similarly, we prove that non-connective algebraic K-theory is the universal localizing invariant, i.e., the universal functor that moreover satisfies the "Thomason-Trobaugh-Neeman" localization theorem. To prove these results, we construct and study two stable infinity categories of "noncommutative motives"; one associated to additivity and another to localization. In these stable infinity categories, Waldhausen's S. construction corresponds to the suspension functor and connective and non-connective algebraic K-theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K-theory of every scheme, stack, and ring spectrum can be recovered from these categories of noncommutative motives. In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable infinity categories. We also explain in detail the comparison between the infinity categorical version of Waldhausen K-theory and the classical definition. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K-theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.
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