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The unification of Quantum Mechanics and General Relativity remains the primary goal of Theoretical Physics, with string theory appearing as the only plausible unifying scheme. In the present work, in a search of the conceptual foundations…
A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for space-time (or space). The starting point is the observation that entities of this type can be regarded as objects in a category…
In Categorial Topology, given a category (as a "geometric object") we can consider its properties preserved under continuous action (a "deformation") of a comma-propagation operation. However, the Metacategory space, valid for all…
Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made…
We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely…
We give to the categorical theory PR of Primitive Recursion a logically simple, algebraic presentation, via equations between maps, plus one genuine Horner type schema, namely Freyd's uniqueness of the initialised iterated. Free Variables…
Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from…
We undertake a reconstruction of the epistemic significance of research on operational theories in quantum foundations. We suggest that the space of operational theories is analogous to the space of possible worlds employed in the possible…
Using the concepts of category and functor, we provide some insights and prove an intrinsic property of the category ${\bf AprS}$ of approximation spaces and relation-preserving functions, the category ${\bf RCls}$ of rough closure spaces…
A rational pseudo-rotation $f$ of the torus is a homeomorphism homotopic to the identity with a rotation set consisting of a single vector $v$ of rational coordinates. We give a classification for rational pseudo-rotations with an invariant…
We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result for simplicial algebras over these theories, showing…
Computational functionalism about consciousness is often criticized for relying on observer-relative interpretations of physical systems. This paper proposes a mathematical refinement of functionalism that avoids this problem. The central…
This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the category-theoretic version of the classical area of algebraic semantics. The…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…
We discuss the global properties of static, spherically symmetric configurations of a self-gravitating real scalar field $\phi$ in general relativity (GR), scalar-tensor theories (STT) and high-order gravity ($L=f(R)$) in various…
This paper offers suggested improvements to the causal sets program in discrete gravity, which treats spacetime geometry as an emergent manifestation of causal structure at the fundamental scale. This viewpoint, which I refer to as the…
Although foundation models (FMs) claim to be powerful, their generalization ability significantly decreases when faced with distribution shifts, weak supervision, or malicious attacks in the open world. On the other hand, most domain…
The present study attempts to provide a consistent and coherent account of what the world could be like, given the conceptual framework and results of contemporary quantum theory. It is suggested that standard quantum mechanics can, and…
The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every…