Systems and Categories

TL;DR

This paper develops a formal framework for systems and categories, providing axioms, definitions, and proofs to unify concepts in mathematics such as groups, automorphisms, and functors, aiming for a comprehensive mathematical description.

Contribution

It introduces a formal concept of systems and categories, constructs the group of integers via axioms, and proves foundational theorems like Cayley's and Yoneda's within this framework.

Findings

Defined a formal concept of systems and categories

Constructed the group of integers using axioms

Provided proofs of Cayley's and Yoneda's lemmas

Abstract

The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we define a group of automorphisms; we are defining an ordered group of functors with a natural transformation between any two. We give an isomorphism from the group of integers into the group of automorphisms, as guaranteed by Cayley's Theorem. The ultimate aim is to use these definitions and concepts, of system and category, to give a general description of mathematics. The third chapter is dedicated to set theory and we provide a proof of the Yoneda Lemma. This is used to prove Cayley's theorem. We see how this relates to the construction of the integers, before introducing representable functors. After lattices, we devote a chapter to group theory. We conlcude with a brief description of topological systems; we describe them as functors on algebraic categories, with a subset of invariant objects. This new version includes a first description of linear spaces. The section for operations has been rewritten and corrected.