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Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalisation from few examples. We formalise this principle through a…

We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality…

We adapt Goldie's concept of uniform dimensions from module theory over rings to $\Gamma$-monoids. A $\Gamma$-monoid $M$ is said to have uniform dimension $n$ if $n$ is the largest number of pairwise incomparable nonzero $\Gamma$-order…

We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with…

In this article we present a new modelling framework for structured concepts using a category-theoretic generalisation of conceptual spaces, and show how the conceptual representations can be learned automatically from data, using two very…

Here, it is shown that the path-integral representation of any stochastic or deterministic continuous-time dynamical model is a cohomological or Witten-type topological field theory, i.e., a model with global topological supersymmetry…

We show that there is a fast algorithm that embeds hierarchical structures in three-dimensional Minkowski spacetime. The correlation of data ends up purely encoded in the causal structure. Our model relies solely on oriented token pairs --…

Let $T$ be a first-order theory. A correspondence is established between internal covers of models of $T$ and definable groupoids within $T$. We also consider amalgamations of independent diagrams of algebraically closed substructures, and…

We illustrate a physical situation in which topological symmetry, its breakdown, space-time uncertainty principle, and background independence may play an important role in constructing and understanding matrix models. First, we show that…

The bootstrap category in E-theory for C*-algebras over a finite space X is embedded into the homotopy category of certain diagrams of K-module spectra. Therefore it has infinite n-order for every n. The same holds for the bootstrap…

We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to "directed spaces". Directed spaces are locally modelled on manifold diagrams, which are stratifications of…

We develop realizability models of intensional type theory, based on groupoids, wherein realizers themselves carry non-trivial (non-discrete) homotopical structure. In the spirit of realizability, this is intended to formalize a homotopical…

Metacognitive theories provide foundational frameworks for understanding self-regulated learning, yet they lack systematic integration into comprehensive scenario taxonomies capable of guiding AI-enhanced professional development…

We introduce a notion of topological property (T) for \'etale groupoids. This simultaneously generalizes Kazhdan's property (T) for groups and geometric property (T) for coarse spaces. One main goal is to use this property (T) to prove the…

As deep generative models have progressed, recent work has shown them to be capable of memorizing and reproducing training datapoints when deployed. These findings call into question the usability of generative models, especially in light…

Formalized $1$-category theory forms a core component of various libraries of mathematical proofs. However, more sophisticated results in fields from algebraic topology to theoretical physics, where objects have "higher structure," rely on…

We present a new metaphysical framework for physics that is conceptually clear, ontologically parsimonious, and empirically adequate. This framework relies on the notion of self-subsisting structure, that is, a set of fundamental physical…

In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it mid ring. Also, we provide new characterizations for von Neumann…

Conditional independence has been widely used in AI, causal inference, machine learning, and statistics. We introduce categoroids, an algebraic structure for characterizing universal properties of conditional independence. Categoroids are…

Scientific studies of consciousness rely on objects whose existence is assumed to be independent of any consciousness. On the contrary, we assume consciousness to be fundamental, and that one of the main features of consciousness is…