
details
Title
Boolean reflections for frames
Abstract The category Frm of frames is an algebraic [pointfree) modification of the category of topological spaces. In particular, each topology is a frame. A study of Frm gives us many algebraic techniques not available in the pointsensitive setting. The category Frm includes the category CBA of complete boolean algebras. At first sight it seems that CBA is a reflective subcategory of Frm, but there is a mysterious settheoretic obstruction. Some
frames can be reflected into CBA and some not. The category Frm is much richer than the category of spaces. In particular, each frame A has an associated larger frame NA, its
assembly, the frame of all nuclei on A. When A is the topology of a space, a nucleus is essentially a Grothendieck topology for the
space. (There are similar gadgets for modules over a ring  the Gabriel topologies for the ring.) The assembly construction N(.) can be iterated through the ordinals
A > NA > N^2A > N^3A > .....
and this tower stabilizes precisely when A has a boolean reflection. Some of the properties of this tower can be measured by an extension of the CantorBendixson process on a topological space. This extension seems to be new for spaces (but does have an analogue for modules). I will explain what I know about this tower, finishing with an example where N^3A is boolean but N^2A is not. Nothing seems to be known beyond this level.
Areas of interest
Topology, Category Theory
Speaker(s)
Harold Simmons (Manchester Univ., UK)
Date
May 16, 2006 Time 16.00 Room
5.5

