This study presents an inter-universal mathematical-logical framework constructed upon the minimal axiom Cogito, ergo sum (CES), integrating the Intermediate Meta-Universe (IMU) and the Hierarchical State Grid (HSG). The CES defines existence as a reflexive correspondence --'to be' and 'to be sayable'--and positions any formal system, including ZFC or HoTT, as an attachable extension atop this minimal structure. The IMU functions as a registry of axiomatic dependencies that connect heterogeneous theories, employing the Institution-theoretic framework to ensure coherent inter-theoretical linkages. The HSG concretizes these ideas through categorical construction, defined by three orthogonal axes: the state-depth axis, the mapping-hierarchy axis, and the temporal axis incorporating the principle of 'no future reference.' Through these, the identity of 'definition = state' is formally established as a categorical property. Extending this structure to biological systems, the neural system is implemented as a 0-3D complex of neuron-function fields on the HSG, while its categorical extensions via fiberization over the material base enable the parallel integration of multiple physiological universes-neural, endocrine, learning, genetic, and input/output systems-into a coherent adjoint ensemble. Within this framework, human behavior and cognition emerge as temporal compositions of inter-universal algorithms constrained by the material base. Finally, by contrasting human cognition, which relies on external CES, with machine existence, this study introduces the concept of internal CES, wherein a machine grounds its own logic upon the factuality of its operation. This internal self-axiomatization establishes a continuous bridge between philosophical ontology and engineering implementation, providing a new foundation for the autonomous and self-defining existence of artificial intelligence.

Previous work has demonstrated that efficient algorithms exist for computing Kan extensions and that some Kan extensions have interesting similarities to various machine learning algorithms. This paper closes the gap by proving that all error minimisation algorithms may be presented as a Kan extension. This result provides a foundation for future work to investigate the optimisation of machine learning algorithms through their presentation as Kan extensions. A corollary of this representation of error-minimising algorithms is a presentation of error from the perspective of lossy and lossless transformations of data.

On the one side, the formalism of Global Transformations comes with the claim of capturing any transformation of space that is local, synchronous and deterministic.The claim has been proven for different classes of models such as mesh refinements from computer graphics, Lindenmayer systems from morphogenesis modeling and cellular automata from biological, physical and parallel computation modeling.The Global Transformation formalism achieves this by using category theory for its genericity, and more precisely the notion of Kan extension to determine the global behaviors based on the local ones.On the other side, Causal Graph Dynamics describe the transformation of port graphs in a synchronous and deterministic way and has not yet being tackled.In this paper, we show the precise sense in which the claim of Global Transformations holds for them as well.This is done by showing different ways in which they can be expressed as Kan extensions, each of them highlighting different features of Causal Graph Dynamics.Along the way, this work uncovers the interesting class of Monotonic Causal Graph Dynamics and their universality among General Causal Graph Dynamics.

A common problem in data science is "use this function defined over this small set to generate predictions over that larger set." Extrapolation, interpolation, statistical inference and forecasting all reduce to this problem. The Kan extension is a powerful tool in category theory that generalizes this notion. In this work we explore several applications of Kan extensions to data science. We begin by deriving a simple classification algorithm as a Kan extension and experimenting with this algorithm on real data. Next, we use the Kan extension to derive a procedure for learning clustering algorithms from labels and explore the performance of this procedure on real data. We then investigate how Kan extensions can be used to learn a general mapping from datasets of labeled examples to functions and to approximate a complex function with a simpler one.

We propose Universal Causality, an overarching framework based on category theory that defines the universal property that underlies causal inference independent of the underlying representational formalism used. More formally, universal causal models are defined as categories consisting of objects and morphisms between them representing causal influences, as well as structures for carrying out interventions (experiments) and evaluating their outcomes (observations). Functors map between categories, and natural transformations map between a pair of functors across the same two categories. Abstract causal diagrams in our framework are built using universal constructions from category theory, including the limit or co-limit of an abstract causal diagram, or more generally, the Kan extension. We present two foundational results in universal causal inference. The first result, called the Universal Causality Theorem (UCT), pertains to the universality of diagrams, which are viewed as functors mapping both objects and relationships from an indexing category of abstract causal diagrams to an actual causal model whose nodes are labeled by random variables, and edges represent functional or probabilistic relationships. UCT states that any causal inference can be represented in a canonical way as the co-limit of an abstract causal diagram of representable objects. UCT follows from a basic result in the theory of sheaves. The second result, the Causal Reproducing Property (CRP), states that any causal influence of a object X on another object Y is representable as a natural transformation between two abstract causal diagrams. CRP follows from the Yoneda Lemma, one of the deepest results in category theory. The CRP property is analogous to the reproducing property in Reproducing Kernel Hilbert Spaces that served as the foundation for kernel methods in machine learning.