In this introductory chapter we explain, in largely non-technical terms, not how monoids and their actions occur everywhere in algebra, but also how they provide a common framework for the ordered, metric, topological, or similar structures targeted in this book. This framework is categorical, both at a micro level, since individual spaces may be viewed as generalized small categories, and at a macro level, as we are providing a common setting and theory for the categories of all ordered sets, all metric spaces, and all topological spaces – and many other categories.

Whilst this Introduction uses some basic categorical terms, we actually provide all required categorical language and theory in Chapter II, along with the basic terms about order, metric, and topology, before we embark on presenting the common setting for our target categories. Many readers may therefore want to jump directly to Chapter III, using the Introduction just for motivation and Chapter II as a reference for terminology and notation.

The ubiquity of monoids and their actions

Nothing seems to be more benign in algebra than the notion of monoid, i.e. of a set M that comes with an associative binary operation m : M × M → M and a neutral element, written as a nullary operation e : 1 → M. If mentioned at all, normally the notion finds its way into an algebra course only as a brief precursor to the segment on group theory.