On completeness in a non-Archimedean setting via firm reflections

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. Br\"ummer 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.