Cometic functors for small concrete categories and an application

TL;DR

This paper introduces cometic functors for small concrete categories, providing a method to derive injective and surjective maps, with an application demonstrating their utility in mathematical contexts.

Contribution

It defines a new functor and natural transformation for small concrete categories, enabling the derivation of injective and surjective maps with an application example.

Findings

F-image of monomorphisms are injective maps

Components of the natural transformation are surjective

Application demonstrates practical use of the functor and transformation

Abstract

Our goal is to derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics. Let A be a small concrete category. We define a functor F, cometic functor, from A to the category Set and a natural transformation \pi, called cometic projection, from F to the inclusion functor of A into Set such that the F-image of every monomorphism A is an injective map and the components of \pi are surjective maps. Also, we give a nontrivial application of F and \pi.