## Mark V Lawson: two tutorial lectures and background material

I am scheduled to give two tutorial lectures. Below you will find drafts of the lectures as well as more detailed supporting notes.

§

Prologue 1

Lecture 1  Sunday at 10.30. A primer on inverse semigroups (Draft).

In this talk, I shall explain how inverse semigroups arose as the abstract versions of pseudogroups of transformations, in much the same way that groups arose as the abstract versions of transformation groups. I shall also explain in what way inverse semigroups can be viewed as extensions of presheaves of groups by pseudogroups. In fact, the lecture will centre on two representations: the Wagner-Preston representation and the Munn representation. I shall also describe the way in which inverse semigroups can be naturally viewed as special kinds of ordered groupoids. To understand this talk, you just need to know what a semigroup is and how homomorphisms are represented by congruences.

Chapter 1 These are the detailed notes that prove all the assertions made in the lecture.

Monday at 10.00. Building inverse semigroups from categories (Draft).

Inverse semigroups can be described in terms of categories in at least two ways. The first, touched on in lecture 1, is as special kinds of ordered groupoids. In this lecture, I shall make the ordered groupoid approach precise, and then describe a different way of using categories to describe inverse semigroups. This originated, unlikely as it might seem, in Girard's work in linear logic and a paper on certain kinds of groups by Patrick Dehornoy. To understand this talk, you just need to know the most basic category theory, and it might help to be aware that I always treat categories as 'monoids with many identities'.

Chapter 2 These are the detailed notes that prove all the assertions made in the lecture.

§