A Category of the Political \Large{Part I - Homónoia}
Joseph Abdou
TL;DR
This work constructs a two-level, category-theoretic model of political organization and transformation by introducing Political Configuration (p-formations) and Political Foundation (p-sites) anchored on a finite Base $I$ and Ground $A$. The Knit and Nerve functors relate Foundation to Configuration, with Canon providing an inverse on suitable domains, thereby unifying structure and interpretation of political dynamics. Recomposition of formations is formalized through BC- and SC-maps, yielding categories $ extbf{P}^2_ullet$ and $ extbf{SIM}_ullet$, while foundation-wide morphisms (B- and S-maps) give rise to two parallel theories $ extbf{bpol}_ullet$ and $ extbf{spol}_ullet$ that are shown to be equivalent to the configuration categories. The framework clarifies how observable political configurations arise from underlying profiles and how transformations at the foundation level induce changes in the configuration, with canonical constructions and subcanonical/simple p-sites providing tractable canonicalizations. Overall, the model offers a rigorous, abstract arena to study political transformation, while recognizing its dependence on external institutional rules and strategic behavior for concrete outcomes. The Macron transition example illustrates how the theory accommodates multi-valued morphisms and coalition realignments, motivating future extensions to capture more complex political dynamics.
Abstract
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.