Stable and Unstable Homotopy Theory of Higher Geometric Stacks

在这个项目中，我们在代数和拓扑两种情况下，研究了基上高几何堆叠的稳定和不稳定同伦理论。这是代数环境下的动机同伦理论和拓扑环境下的等变同伦理论的推广。我们将不稳定同伦范畴构造为Nisnevich拓扑中束的∞范畴的a1 -局部化。然后构造稳定同伦范畴为这些∞范畴相对于基栈上与向量束相关联的球束的固定集合的镇定性。我们建立了这些∞-范畴的基本结果，如同局部纯度。作为一种重要的计算工具，我们证明了光滑和固有映射的对偶定理。在代数设置中，当基是线性约化代数群的分类堆栈时，会出现一个特别重要的情况。然后我们的构造提供了一个到目前为止只考虑有限群的等变动力同伦理论。研究了具有G作用的二次型Grothendieck-Witt群在有限群G情况下的等变动力球谱的自同态环。虽然在拓扑设置中已经理解了紧李群的类似情况，但我们的理论允许更高的群作用，例如弦群或环群。我们提出了一个更高的tom Dieck分裂猜想，并研究了它与Waldhausen a -理论函子的关系。

In this project we undertake a study of the stable and unstable homotopy theory of higher geometric stacks over a base, in both the algebraic and topological settings. This is a generalization of motivic homotopy theory in the algebraic setting and of equivariant homotopy theory in the topological setting. We construct the unstable homotopy category as the A1-localization of the ∞-category of sheaves in the Nisnevich topology. The stable homotopy category is then constructed as the stabilization of these ∞-categories with respect to a fixed set of sphere bundles associated to vector bundles on the base stack. We establish foundational results, such as homotopical purity, for these ∞-categories. As an important calculational tool we prove a duality theorem in this setting for smooth and proper maps. In the algebraic setting a particularly important case occurs when the base is the classifying stack of a linearly reductive algebraic group. Our construction then provides an equivariant motivic homotopy theory, which has thus far only been considered for finite groups. We study the endomorphism ring of the equivariant motivic sphere spectrum which in the case of a finite group G is related to the Grothendieck-Witt group of quadratic forms with G-action. While the analogous case of a compact Lie group is already understood in the topological setting, our theory allows for higher group actions such as string groups or loop groups, for example. We formulate a higher tom Dieck splitting conjecture and study the relationship with Waldhausen's A-theory functor.