Recent advances show that certain branches of category theory can be used to capture the essential behavioural properties of many complex systems, and provides the right language to study their foundational concepts across a broad range of disciplines. In particular: Monoidal categories provide the quantum mechanical formalism with an intuitive diagrammatic language, which meanwhile has been adopted by many researchers in the quantum foundations community. They provide essential new insights in the foundations of physics by giving the manner in which systems compose a privileged role. Closely related formalisms also provide a deep understanding of how words in natural language interact to produce the meaning of a sentence, and underpin the mechanisms of Bayesian updating. Sheaf theory gives decisive insights into the structure of quantum contextuality, including non-locality, as part of a general ‘logic of contextuality’ with wideranging applications to modelling physical, informational and linguistic processes. In mathematics it describes the relationship between local and global properties, and connects geometry and logic. Sheaf structures form the basis for proposals of a theory of quantum gravity. Coalgebra provides a powerful set of tools for modelling dynamical systems, encompassing reflexive and non-well-founded aspects; systems which can exhibit self-reference and evolve by interaction with their environments. The aim of this project is to explore the connections between these approaches, to combine them into a unified formalism for modelling a wide range of complex systems, and to explore its conceptual significance. We will investigate how this structure arises in different areas. Ultimately, we will explore how this combined structure could lead to a unique foundational fully integrated formalism for modeling the physical world, making deductions about it, speaking about it, and relating it to other features and concepts in our reality.