发表机构
University of Aberdeen; Universität Münster（阿伯丁大学; 明斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于CIMPA讲座，证明泛函分析、算子代数及稳定范畴中的若干范畴可对偶，由此定义解析空间代数K-理论，并类比拓扑与代数K-理论，关联装配映射猜想。
AI 中文摘要
本文基于2025年CIMPA算子代数与K-理论学校系列讲座，强调利用可对偶范畴研究非交换几何的现代视角。为此，我们证明泛函分析（bornological与condensed模的导出范畴）、算子代数（E-理论）及稳定范畴（非交换动机）中自然出现的若干范畴是可对偶的。这尤其为解析空间代数K-理论的定义提供了途径。此外，E-理论与动机的可对偶性促使我们探索拓扑K-理论与代数K-理论之间的类比，其灵感源于装配映射（Baum-Connes与Farrell-Jones）猜想。
英文摘要
In this exposition based on a lecture series at the CIMPA School on Operator Algebras and K-theory, 2025, we highlight modern perspectives on noncommutative geometry using dualisable categories. To this end, we show that several categories naturally arising in functional analysis (derived categories of bornological and condensed modules), operator algebras (E-theory) and stable categories (noncommutative motives) are dualisable. This in particular leads to a definition of algebraic K-theory for analytic spaces. Furthermore, the dualisability of E-theory and motives leads us to pursue an analogy between topological and algebraic K-theory, inspired by assembly map (Baum-Connes and Farrell-Jones) conjectures.
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