Related papers: Systems and Categories
Written to be contributed as the "mathematical modeling" chapter of a book, edited by Elaine Landry, to be titled "Categories for the Working Philosopher". In this chapter, category theory is presented as a mathematical modeling framework…
Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of…
Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in…
This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…
`Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
This book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It gives an extensive and precise coverage of the subjects which are addressed, in a consistent and…
Based on Gandy's principles for models of computation we give category-theoretic axioms describing locally deterministic updates to finite objects. Rather than fixing a particular category of states, we describe what properties such a…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
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…
This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important…
This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on…
This article presents a general description of dynamical systems using the language of enriched functors and enriched natural transformations. This framework is essential to establish the equivalence of three descriptions of dynamics -- a…
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 provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
The purpose of this book is to lay out certain aspects of descriptive set theory. After initially establishing notation and generalities we proceed to the following topics: partitions, semirings, rings, $\sigma$-rings, $\delta$-rings,…
This paper provides a complete suite of axioms for a version of set theory that I call Explication. Explication borrows from the two most prominent existing systems of set theory. Explication starts with class variables. After several…