适用于外同伦与上同调理论的稳定框架
A stable framework for proper and exterior homotopy and cohomology theories
AI总结：
该研究构建了外同伦理论的稳定框架，将其应用于恰当同伦理论，建立了约化外上同调理论的Brown可表示性定理，为恰当同伦理论的稳定理论发展提供了关键基础。
AI中文摘要：
我们构建了一个外同伦理论的稳定框架，并将其应用于恰当同伦理论。从外空间范畴上已知的单纯模型结构出发，我们过渡到以射线为基的可回缩外空间的恰当胞腔模型范畴，建立了对应的外谱稳定同伦理论。该框架提供了稳定同伦理论中若干经典构造的外版本，并为恰当同伦理论带来新视角，而恰当同伦理论因缺乏合适的范畴性质，长期以来难以发展出令人满意的稳定理论。作为应用，我们建立了约化外上同调理论的Brown可表示性定理，这一结果给出了源自外框架的恰当上同调理论的可表示性结论，包括与经典广义上同调理论相关的紧支集上同调理论。
英文摘要：
We develop a stable framework for exterior homotopy theory and apply it to proper homotopy theory. Starting from a known simplicial model structure on the category of exterior spaces we then pass to the proper, cellular model category of retractive exterior spaces over the ray, and establish the associated stable homotopy theory of exterior spectra. The resulting framework provides exterior analogues of several classical constructions in stable homotopy theory and sheds new light on proper homotopy theory, where the development of a satisfactory stable theory has been hindered by the lack of suitable categorical properties. As an application, we establish a Brown representability theorem for reduced exterior cohomology theories. This yields representability results for proper cohomology theories arising from the exterior setting, including cohomology theories with compact supports associated with classical generalized cohomology theories.
