Categorical Frameworks for AI: A Survey and a TPN-based Foundation for Operational and Semantic Analysis

Modern AI systems, such as deep learning models, remain opaque and lack unified theoretical foundations.Category theory offers a mathematical language to integrate diverse paradigms of AI: gradient-based, probabilistic,and geometric learning, but a gap persists between abstract theory and concrete computation.We introduce Typed Petri Nets (TPNs) as a formal and operational framework bridging this gap. TPNs modelneural architectures compositionally and unify deterministic and probabilistic data flows. By linking TPNs with sheafand cosheaf theory, we formalize the relation between local computation and global consistency, and by derivingcausal nets from TPN executions, we analyze causal dependencies within AI processes. This framework advances theinterpretability and verifiability of AI systems.