Almost all algebraic structure in common use have some sort of associative binary operation ("multiplication"), but we don't always require there to exist all, or even any, inverses with respect to the operation. For instance, the definition of a group and a monoid are identical except that in a group every element has an inverse, whereas in a monoid this need not be the case.
However, through a procedure called 'localisation', we can add inverses constructively (thereby turning e.g. a monoid into a group). This type of construction can be applied to other algebraic structures, such as rings, modules, and categories.
V. F. R. Jones has described how forming the localisation of an algebraic structure automatically also gives 'localisations' for representations of (i.e. functors from) that algebraic structure.
I'm going to explain how this can be used to construct discrete toy models in physics using representations of Thompson's groups. (There're called *toy* models because our spacetime is [probably] not discrete.) I'll also talk about some fascinating purely mathematical questions around Thompson's groups.
I will assume zero knowledge of any of the physics involved – everything will be explained and illustrated.