The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms

TL;DR

This paper introduces the Quillen negation monoid for categories, explores its action on topological space morphism classes, and computes the Schreier graph for a key class, linking topology and category theory.

Contribution

It defines the Quillen negation monoid as a faithful quotient of a free monoid acting on morphism classes and explicitly computes its Schreier graph in the category of topological spaces.

Findings

Finiteness of the orbit of the class {∅→*} with 21 classes

Explicit description of classes like maps with sections, dense images, and quotients

Connection of orbit classes to topological properties like connectedness and separation axioms

Abstract

The free monoid with two generators acts on classes (=properties) of morphisms of a category by taking the left or right orthogonal complement with respect to the lifting property, and we define the Quillen negation monoid of the category to be its largest quotient which acts faithfully. We consider the category of topological spaces and show that a number of natural properties of continuous maps are obtained by applying this action to a single example. Namely, for the category of topological spaces we show finiteness of the orbit of the simplest class of morphisms { \emptyset \to {*} }, and we calculate its Schreier graph. The orbit consists of 21 classes of morphisms, and most of these classes are explicitly defined by standard terminology from a typical first year course of topology: a map having a section or dense image; quotient and induced topology; surjective, injective; (maps representing) subsets, closed subsets; disjoint union, disjoint union with a discrete space; each fibre satisfying separation axiom T0 or T1 . Also, the notions of being connected, having a generic point, and being a complete lattice, can be defined in terms of the classes in the orbit. In particular, calculating parts of this orbit can be used in an introductory course as exercises connecting basic definitions in topology and category theory.