Plots and Their Applications - Part I: Foundations

TL;DR

This paper extends category theory to a broader setting called plots, introducing new concepts and structures that unify various mathematical frameworks, laying groundwork for future theories of normed and measure spaces.

Contribution

It formulates an abstract, identity-free framework for plots, generalizing categories and related structures, and introduces new notions like regular representations and punctors.

Findings

Defines a relaxed composition law for plots.

Introduces an identity-free approach to isomorphisms and limits.

Lays foundation for a unified theory of normed and measure spaces.

Abstract

The primary goal of this paper is to abstract notions, results and constructions from the theory of categories to the broader setting of plots. Loosely speaking, a plot can be thought of as a non-associative non-unital category with a "relaxed" composition law: Besides categories, this includes as a special case graphs and neocategories in the sense of Ehresmann, Gabriel's quivers, Mitchell's semicategories, and composition graphs, precategories and semicategories in the sense of Schr\"oder. Among other things, we formulate an "identity-free" definition of isomorphisms, equivalences, and limits, for which we introduce regular representations, punctors, $\mathcal M$-connections, and $\mathcal M$-factorizations. Part of the material will be used in subsequent work to lay the foundation for an abstract theory of "normed structures" serving as a unifying framework for the development of fundamental aspects of the theory of normed spaces, normed groups, etc., on the one hand, and measure spaces, perhaps surprisingly, on the other.