Systems and Categories
Juan Pablo Ramirez

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…
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Taxonomy
TopicsLogic, programming, and type systems
