AI 中文总结
研究满足交换定理的两个函子的关系，通过确定泛范畴及相关扭曲实现其典范关联，还将结果应用于三函子形式体系，给出“相干六个运算”的1 - 范畴实现，证明通用性时开发了计算范畴的技术。
AI 中文摘要
我们定义了交换定理的概念，并表明满足交换定理的任意两个函子通过扭曲范数映射具有典范关系。通过确定接收一对满足交换定理的函子的泛范畴来实现这一点。此外，我们表明所出现的扭曲本质上是K - 理论的，由虚拟向量丛的范畴化类似物参数化。作为应用，我们表明每个三函子形式体系都有一个典范扩展，它编码了该形式体系内部的庞加莱对偶性和托姆扭曲。这给出了Hoyois所概述的“相干六个运算”的1 - 范畴实现。在证明通用性的过程中，开发了用于计算与双和n - 单纯空间相关的范畴的技术。这个方向上的许多结果可被视为Liu - Zheng工作的与模型无关的重新推导。
英文摘要
We define the notion of an exchange theorem and show that any two functors satisfying an exchange theorem are canonically related via twisted norm maps. This is done by identifying the universal category receiving a pair of functors satisfying an exchange theorem. Additionally, we show that the twists occurring are $K$-theoretic in nature, parametrized by a categorified analogue of virtual vector bundles. As an application, we show that every 3-functor formalism has a canonical extension which encodes Poincaré duality and Thom twists internal to the formalism. This gives a 1-categorical realization of the "coherent six operations" outlined by Hoyois. In the process of proving universality, techniques for computing categories associated to bi- and $n$-simplicial spaces are developed. Many of the results in this direction may be viewed as model-independent rederivations of work of Liu--Zheng.
Comments82 pages
