Uniform environments as a general framework for metrics and uniformities

This paper puts theories of classical uniformity, probabilistic uniformity, Hutton uniformity, classical metric, probabilistic metric, and Hutton metric into one framework by introducing the concept of uniform environment, which is a functor from the category of sets to the opposite category of involutive, unital quantales and involutive, closed unital maps subject to certain conditions. The results show that uniform environments and their transformations provide a nice framework to study relationships between various notions of uniformity in the literature.