Causal-net category

Abstract

A causal-net is a finite acyclic directed graph. In this paper, we introduce a category, denoted as Cau, whose objects are causal-nets and morphisms are functors of path categories of causal-nets. It is called causal-net category and in fact the Kleisli category of the "free category on a causal-net" monad. We study several composition-closed classes of morphisms in Cau, which characterize interesting causal-net relations, such as coarse-graining, contraction, immersion-minor, topological minor, etc., and prove several useful decomposition theorems. In addition, we show that the notions of a coloring and a minor can be understood as a special kind of minimal-quotient and sub-quotient in Cau, respectively. Base on these results, we conclude that Cau is a natural setting for studying causal-nets, and the theory of Cau should shed new light on the category-theoretic understanding of graph theory.