Completeness in Generalized Ultrametric Spaces

Γ-ultrametric spaces are spaces which satisfy all the axioms of an ultrametric space except that the distance function takes values in a complete lattice Γ instead of R≥0. Γ-ultrametric spaces have been extensively studied as a way to weaken the notion of an ultrametric space while still providing enough structure to be useful (see for example [17], [18], [8]). The many uses of Γ-ultrametric spaces include being a valuable context to study equivalence relations ([19]), trees ([2]), domain theory ([12]), and logic programming ([? ], [9], [21]). The goal of this paper is to characterize the relationship between four important notions of completeness for Γ-ultrametric spaces: spherical completeness, Cauchy completeness, strong Cauchy completeness and injectivity. We begin in Section 2.1 by reviewing the de nition of a Γ-ultrametric space and of related notions such as the closed ball functor, γ-Cauchy nets and γ-Cauchy functions. In Section 2.2 we review the de nition of spherical completeness and give two characterizations of spherical completeness in terms of γ-Cauchy nets. Finally in Section 2.3 we give the de nition of Cauchy completeness, strong Cauchy completeness and injectivity. We also give a characterization of strong Cauchy completeness in terms of the closed ball functor, in terms of spherical completeness, as well as showing that the notion of strong Cauchy completeness coincides with injectivity for Γ-ultrametric spaces. In Section 3 we will consider the notion of a completion of a Γ-ultrametric space. In Section 3.1 we will show that the category of Γ-ultrametric spaces is equivalent to the category of skeletal symmetric Γ-enriched categories. We then use this equivalence to show that every Γ-ultrametric space has a Cauchy completion. In Section 3.2 we show not all Γ-ultrametric spaces have a strong Cauchy completions even though each Γ-ultrametric space does have a minimal strong Cauchy complete extension. In the process we will show, in Section 3.2.1, that the category of Γ-ultrametric spaces is equivalent to the category of abby separated presheaves on Γ.