Workshop Programme

9:00   am -- Welcome: Ernie Manes, Robin Cockett (45 sec. or less)

9:05   am -- Mark Kambites
                     A gentle introduction to inverse semigroups

10:05 am -- John Fountain
                     Ample semigroups a survey

11:05 am -- Ben Steinberg
                    Finite categories and the decomposition theory of finite monoids

12:00 Lunch

1:00   pm -- Ernie Manes
                     Restrictions categories and guards

2:00   pm -- Steve Lack
                     Restriction categories and partial products

3:00   pm -- Pedro Resende 
                     From groupoids and inverse semigroups to supported quantales

4:00 Tea

4:15   pm -- Robin Cockett
                     Topological aspects of restriction categories

           Open discussion: are we going different places? Other directions.


(1) Mark Kambites:
We give a gentle introduction to the theory of inverse semigroups, and explain how categories and groupoids arise in their study.

(2) John Fountain:
Abstract here.

(3) Ben Steinberg:
In the late 80s, B. Tilson introduced the notion of the derived category of a relational morphism of finite monoids as a replacement for the group kernel in finding wreath product decompositions.  He also introduced the theory of pseudovarieties of categories.  Since then the use of small categories as generalized monoids has played a key role in the theory of monoids.  These results have also been generalized by various authors to wreath product (or Grothendeick construction) decompositions of categories, which in turn leads to new results about semigroups.  

(4) Ernie Manes:
 Has a handout!

(5) Steve Lack:
I will describe the sense in which restriction categories are abstract categories of partial maps (in particular they are full subcategories of categories of partial maps). I  will also compare them with some other treatments of partial maps, notably those where there is a notion of product (the cartesian product in the world of total maps).

(6) Pedro Resende:
There are various known constructions of groupoids from inverse semigroups, such as the ordered groupoid of an inverse semigroup, the germ groupoid of a pseudogroup, Paterson's universal groupoid of an inverse semigroup, etc. In the first part of this lecture I shall describe a general topological correspondence that subsumes many of these constructions, namely a (non-functorial) `duality' between topological \'{e}tale groupoids and complete infinitely distributive inverse semigroups. The correspondence is based on seeing such semigroups as being sheaves, and it can be regarded as a generalization of the fundamental equivalence between sheaves and local homeomorphisms. In the second part of the lecture I shall address an even more general three-fold correspondence involving quantales, which extends the previous constructions in a purely algebraic (and constructive) way; in particular, topological spaces can be replaced by locales. The quantales that arise in these constructions are of a kind known as supported, and I will conclude with an overview of their algebraic properties, including general relations to inverse semigroups and guarded semigroups.

(7) Robin Cockett:
All restriction categories have a "fundamental functor" into the opposite of category of semilattices with stable maps.  Even though this category of semilattices is a very poor cousin of the category locales, it nonetheless has enough structure to support some very basic topological notions.  For example the notion of an open maps can be defined in this setting.  Encouragingly a restriction category has its fundamental functor landing in the the subcategory of these open maps if and only if it is a range restriction category

What about restriction categories whose fundamental functor lands in locales?  These are called join restriction categories and here, of course, the interaction with topological notions takes off (in more than one way).   Closed maps allow one to define Haussdorf objects and proper maps allow one to define compact objects.  In a different direction one can also describe manifolds, local homeomorphisms ...

The aim of the talk is to give a brief taste of these ideas ...