通过坚定的反思论非阿基米德环境中的完备性
On completeness in a non-Archimedean setting via firm reflections
摘要：
本文在G.C.L.提出的“对象完备化的范畴概念”的基础上，发展了（一般）非阿基米德空间的完备化理论。Brümmer和E.朱利我们的背景是所有Hausdorff非阿基米德空间和一致连续映射的构造$\mathbf{NA}_0$，$\mathcal{V}$是$\mathbf{NA}_0$中所有满射嵌入的类。我们确定类${\bf Inj} \mathcal{V}$的所有$\mathcal{V}$ -内射对象，我们提出了一个内部表征为“完整的对象”。这个特征的基本工具是一个小集合的概念，在某种意义上保持了非阿基米德结构的包含顺序。我们证明了由所有完备对象组成的完全子结构$\mathbf{CNA}_0$构成一个强$\mathcal{V}$ -反射子范畴。这意味着$\mathbf{NA}_0$中的每个对象$X$都有一个完成，它是$\mathcal{V}$ -将$r_X:X\to RX$反射到“完全空间”的完整子结构$\mathbf{CNA}_0$中。此外，这种完全化是唯一的（直到同构），在这个意义上，考虑到$L(\mathbf{CNA}_0)$，所有那些态射$u: X\to Y$的类，$Ru:RX\to RY$是一个同构，人们有$\mathcal{V}$包含在$L(\mathbf{CNA}_0)$中。事实上，一个甚至有$\mathcal{V}=L(\mathbf{CNA}_0)$。最后，我们将我们的构造应用到Hausdorff非阿基米德一致空间的经典情形，在这种情况下，我们的完备化约为标准完备化。
英文摘要：
We develop a completion theory for (general) non-Archimedean spaces based on the theory on "a categorical concept of completion of objects" as introduced by G.C.L. Brümmer and E. Giuli. Our context is the construct $\mathbf{NA}_0$ of all Hausdorff non-Archimedean spaces and uniformly continuous maps and $\mathcal{V}$ is the class of all epimorphic embeddings in $\mathbf{NA}_0$. We determine the class ${\bf Inj} \mathcal{V}$ of all $\mathcal{V}$-injective objects and we present an internal characterization as "complete objects". The basic tool for this characterization is a notion of small collections that in some sense preserve the inclusion order on the non-Archimedean structure. We prove that the full subconstruct $\mathbf{CNA}_0$ consisting of all complete objects forms a firmly $\mathcal{V}$-reflective subcategory. This means that every object $X$ in $\mathbf{NA}_0$ has a completion which is a $\mathcal{V}$-reflection $r_X:X\to RX$ into the full subconstruct $\mathbf{CNA}_0$ of "complete spaces". Moreover this completion is unique (up to isomorphism) in the sense that, considering $L(\mathbf{CNA}_0)$, the class of all those morphisms $u: X\to Y$ for which $Ru:RX\to RY$ is an isomorphism, one has that $\mathcal{V}$ is contained in $L(\mathbf{CNA}_0)$. In fact one even has $\mathcal{V}=L(\mathbf{CNA}_0)$. Finally we apply our constructions to the classical case of Hausdorff non-Archimedean uniform spaces, in that case our completion reduces to the standard one.
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