微分拓扑空间之间的广义映射
Generalized maps between diffeological spaces
AI总结：
本文借鉴 Colombeau 广义函数提出微分拓扑空间间的渐近映射，构造具有完备性与余完备性的浓化范畴，并用以建立光滑相对胞腔复形的同伦扩张性质。
摘要：
通过运用 Colombeau 广义函数的思想，我们引入了任意微分拓扑空间之间渐近映射的概念。由微分拓扑空间和渐近映射组成的范畴在微分拓扑空间范畴上是浓化的，并继承了完备性与余完备性。特别地，欧氏开集上的渐近函数集合包含 Schwartz 分布，并在 Robinson 渐近数域上构成一个 Colombeau 型光滑微分代数。为说明我们这套机制的有效性，我们证明：如果利用渐近映射而非光滑映射，则可以对光滑相对胞腔复形建立同伦扩张性质。
英文摘要：
By utilizing the idea of Colombeau's generalized function, we introduce a notion of asymptotic map between arbitrary diffeological spaces. The category consisting of diffeological spaces and asymptotic maps is enriched over the category of diffeological spaces, and inherits completeness and cocompleteness. In particular, the set of asymptotic functions on a Euclidean open set include Schwartz distributions and form a Colombeau type smooth differential algebra over Robinson's field of asymptotic numbers. To illustrate the usefulness of our machinery, we show that homotopy extension property can be established for smooth relative cell complexes if we exploit asymptotic maps instead of smooth ones.
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