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.
Abstracts:
(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 ...