Stable and Unstable Homotopy Theory of Higher Geometric Stacks

在这个项目中，我们从代数和拓扑两个方面研究了基上高几何堆栈的稳定和不稳定同伦理论。这是动机同伦理论在代数环境下的推广，是等变同伦理论在拓扑环境下的推广。我们将不稳定同伦范畴构造为∞-范畴在Nisnevich拓扑中的A1-局部化。然后，稳定同伦范畴被构造为这些∞-范畴相对于与基栈上的向量丛相关联的球丛的固定集合的稳定化。我们建立了这些∞-范畴的基础性结果，例如同伦纯性。作为一个重要的计算工具，我们在这一背景下证明了光滑真映射的一个对偶定理。在代数环境中，当基是线性约化代数群的分类堆栈时，会出现一种特别重要的情况。然后，我们的构造提供了一个等变的动机同伦理论，到目前为止，它只被考虑用于有限群。研究了有限群G的等变运动球谱的自同态环，它与具有G作用的二次型的Grothendieck-Witt群有关。虽然紧致李群的类似情况在拓扑环境中已经被理解，但我们的理论允许更高的群作用，例如弦群或环群。我们建立了一个高阶Tom Dieck分裂猜想，并研究了它与Waldhausen的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.