Categorical and Topos-Theoretic Foundations for Mathematical Modelling of Complex Adaptive Systems in African Contexts

This dissertation develops a categorical and topos-theoretic foundational framework for the mathematical modelling of complex adaptive systems, with original contributions to the structural mathematics of adaptive computation and an application domain centred on the modelling of informal economic and social adaptive networks in West Africa, including the Nigerian informal sector. Category theory and topos theory offer the most general available language for describing structural relationships between mathematical objects, and their application to complex adaptive systems modelling provides a level of abstraction that subsumes both classical differential equation models and discrete computational models within a single coherent framework. The dissertation develops an original doctrine of adaptive-system morphisms in the bicategorical setting, formalising the notion of adaptive equivalence between systems with different internal state spaces but functionally identical macroscopic behaviour. A Grothendieck topos of adaptive system sheaves is constructed, and the internal logic of this topos is developed as a constructive type theory capable of expressing adaptive behavioural laws without committing to a fixed external ontology. The dissertation proves novel adjunction theorems relating the topos of adaptive system sheaves to the topos of temporal sets, establishing a precise correspondence between temporal evolution and adaptive structural change. Applications to the mathematical modelling of the Nigerian informal credit network, the rotating savings and credit association system, and the informal apprenticeship system are developed as concrete case studies within the categorical framework. Keywords: category theory, topos theory, complex adaptive systems, informal economy, categorical foundations