自由单纯形原幂零与pro-$p$分解的连续比较
Continuous Comparison of Free Simplicial Prounipotent and Pro-$p$ Resolutions
浏览论文内容
中文总结 AI 辅助
本文给出原幂零与pro-$p$群范畴中自由单纯形分解比较定理的连续版本，通过连续线性分裂或投射性实现提升，并确立同伦唯一性。
中文摘要 AI 辅助
我们给出了特征零域上的原幂零群范畴和pro-$p$群范畴中自由单纯形分解的比较定理的连续版本。这些分解带有与退化性相容的自由基；其秩可能无限。在特征零情形下，提升步骤由连续线性分裂和完备自由李代数的泛性质得出。在pro-$p$情形下，它由自由pro-$p$群的经典投射性得出。相对提升论证和一个显式单纯形柱面确立了同伦唯一性。我们使用单纯形范畴中的实际满射来表述自由基，因此由单纯形恒等式关联的退化词从一开始就被识别，并且我们详细列出了提升论证中使用的完整匹配对象。该问题在我们先前关于子可收缩表示的Bousfield--Kan完备化的工作中自然产生。
英文摘要
We give a continuous version of the comparison theorem for free simplicial resolutions in the categories of prounipotent groups over a field of characteristic zero and of pro-$p$ groups. The resolutions carry free bases compatible with degeneracies; their ranks may be infinite. In characteristic zero the lifting step follows from continuous linear splittings and the universal property of completed free Lie algebras. In the pro-$p$ case it follows from the classical projectivity of free pro-$p$ groups. A relative lifting argument and an explicit simplicial cylinder establish homotopy uniqueness. We formulate the free bases using actual surjections in the simplex category, so degeneracy words related by simplicial identities are identified from the outset, and we spell out the full matching objects used in the lifting argument. The question arose naturally in our preceding work on Bousfield--Kan completions of subcontractible presentations.
