Diagrams of quantales and Lipschitz norms

Abstract

It is well known that metric spaces are an instance of categorical enrichment in a particular quantale. We show that in a categorically natural way a notion of Lipschitz norm arises in the context of an arbitrary diagram of quantales, instead of just one particular quantale. The generalised Lipschitz norm we present depends functorially on the diagram and is itself a functor to the indexing category of the diagram. The entire process is, in a way we make precise, an instance of a concrete Grothendieck construction.