Toward a category of the political Part I-Homónoia

This is the first part of a work that aims at providing a model of political intertwining and transformation. Formally we introduce the Mathematical category of Political Foundation, which is based on two variables : one variable, called the society, consists of a finite number of parties, while the other, called the ground, consists of a set of states that reflect all relevant interests/values of the society. An object of the category, called a political site, describes how the interests/values of the society are intertwined between the parties. A morphism between political sites consists of a pair of maps, namely a society map and a ground map, satisfying appropriate conditions. In fact we define two versions of the Political Foundation, Bpol and Spol, with the same objects, corresponding to two types of maps, Bmaps and S-maps. We introduce also a second category referred to as the Political Configuration, based on one variable, the society. An object of this category called a complex extends the notion of simplicial complex, and a morphism extends the well-known notion of simplicial map. Two functors that relate the Foundation and the Configuration, the Knit and the Nerve are considered. It turns out that the Knit reveals the structure of Bpol and the Nerve reveals that of Spol.