A Category of the Political \Large{Part I - Homónoia}

This research aims at providing a mathematical model of the organization of the polity and its transformation. For that purpose we construct two categories named respectively Political Configuration and Political Foundation. Our construction depends on a couple of variables called the foundational pair. One variable, called the Base, consists of a finite number of members (agents), while the other, called the Ground, consists of a set of states that reflect all relevant interests/values/aspirations of the base members. An object of the Configuration, called p-formation, extends the notion of simplicial complex, and a morphism, which expresses the recomposition of the base, extends the notion of simplicial map. An object of the Foundation, called p-site, describes the profile of the polity, that is, how the states of the ground are intertwined between the agents. A morphism between political sites consists of a pair of maps, namely a Base map and a Ground map, satisfying appropriate conditions. Two functors relate the Foundation and the Configuration: the Knit which attributes to each p-site a p-formation and the Nerve which attributes to each p-site a simplicial complex. In the opposite direction a functor, called Canon, which attributes to any p-formation its canonical p-site, turns out to be in an appropriate sense the inverse of the Knit and the Nerve.