Category Theory as a Foundation for the Concept Analysis of Complex Systems and Time Series

Wille’s formal concept analysis, Hardegree’s treatment of natural kinds, and Birkhoff’s mathematical theory of polarities provide essentially equivalent tools for the analysis of a static system functioning at a single level. We now show how a number of categorical notions allow these tools to be extended to cover the analysis of complex systems involving multiple hierarchical levels indexed by a semilattice, including the case where a chain represents a time series governing the evolution of a single system. A semilattice is a poset category with finite products or coproducts. Our analysis then rests on functors from a semilattice to the category of complete lattice homomorphisms, or from a semilattice to a category of polarities and bonding relations.