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Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…
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…
The question "What is category theory" is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of…
Conscious experience permeates our daily lives, yet general consensus on a theory of consciousness remains elusive. In the face of such difficulty, an alternative strategy is to address a more general (meta-level) version of the problem for…
A more realistic object detection paradigm, Open-World Object Detection, has arisen increasing research interests in the community recently. A qualified open-world object detector can not only identify objects of known categories, but also…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
In standard first order predicate logic with identity it is usually taken that $a=a$ is a theorem for any term $a$. It is easily shown that this enables the apparent proof of a theorem stating the existence of any entity whatsoever. This…
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…
The concept of category from mathematics happens to be useful to computer programmers in many ways. Unfortunately, all "good" explanations of categories so far have been designed by mathematicians, or at least theoreticians with a strong…
We present an abstract, categorical formulation of dependent functions in a fundamental manner and independently from the Sigma-construction. For that, we define first the notion of a category with family-arrows, or a $\f$-category. A $(\f,…
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…
A new approach is suggested to characterize algebraically automorphisms of the category of free algebras of a given variety. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category…
Category theory can be used to state formulas in First-Order Logic without using set membership. Several notable results in logic such as proof of the continuum hypothesis can be elegantly rewritten in category theory. We propose in this…
Based on the intuitive notion of convexity, we formulate a universal property defining interval objects in a category with finite products. Interval objects are structures corresponding to closed intervals of the real line, but their…
Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
It is well-known that biological phenomena are emergent. Emergent phenomena are quite interesting and amazing. However, they are difficult to be understood. Due to this difficulty, we propose a theory to describe emergence based on a…
Motivated by team semantics and existential second-order logic, we develop a model-theoretic framework for studying second-order objects such as sets and relations. We introduce a notion of abstract elementary team categories that…
The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes…
The notions of null-sets and nullity are present in all discourses of mathematics. They are based on the dual-pair of notions of "almost-every" and "almost none". A notion of nullity corresponds to a choice of subsets that one interprets as…