AI 中文总结
本文引入（李）全局∞-范畴框架，证明等变与全局同伦论具有泛性质，刻画不稳定到稳定同伦论的过渡，定义参数化等变与全局汤姆谱函子并恢复经典构造。
AI 中文摘要
我们引入了（李）全局∞-范畴的框架，该框架形式化了由紧李群索引且配备了沿连续群同态的合适限制函子的各类∞-范畴，这些范畴在等变同伦论与表示论中自然出现。作为主要结果，我们证明在该框架下，不稳定与稳定等变及全局同伦论具有泛性质，细化并推广了arXiv:2301.08240与arXiv:2307.11001中关于有限群的结果。特别地，我们在全局∞-范畴层面刻画了从不稳定到稳定等变及全局同伦论的过渡，其在适当意义下是泛地逆表示球的作用。在此基础上，我们定义了参数化等变与全局汤姆谱函子，并证明它们恢复了基于点集模型定义的经典汤姆谱构造。
英文摘要
We introduce a framework of (Lie-)global $\infty$-categories, which formalizes various families of $\infty$-categories indexed by compact Lie groups and equipped with suitable restriction functors along continuous group homomorphisms that occur naturally in equivariant homotopy theory and representation theory. As our main results, we show that in this framework unstable and stable equivariant and global homotopy theory admit universal properties, refining and generalizing the results for finite groups from arXiv:2301.08240 and arXiv:2307.11001. In particular, we characterize the passage from unstable to stable equivariant and global homotopy theory at the level of global $\infty$-categories as universally inverting the action of representation spheres in an appropriate sense. Building on this, we define parametrized equivariant and global Thom spectrum functors and show that they recover classical Thom spectrum constructions defined in terms of pointset models.
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