 
Summary: Ultrametric Spaces and Presheaves
by Nathanael Leedom Ackerman
Abstract
In this paper we will explore the relationship between Ultrametric
Spaces and Separated Presheaves. We will then use this relationship
to prove some new results about Separated Presheaves with nice prop
erties.
Ultrametric spaces (as in this paper) have been extensively studied (see for
example [7], [8], [1]) as a way to weaken the notion of an ultrametric space
while still providing enough structure to be useful. Ultrametric spaces are
spaces which satisfy all the axioms of an ultrametric space except that the
distance function takes values in a fixed complete lattice (and not necessarily
in the reals) 1
We begin this paper in Section 1 by reviewing the relevant definitions
before discussing in Section 2 the close connection between the categories of
Ultrametric Spaces and the category of separated presheaves on op
.
In Section 3, we will show how all categories with epimonic factoriza
tions can be viewed as having Homsets which are Ultrametric spaces. This
will then allow us, in Section 4, to translate results about Generalized Ul
