Algebra, Logic and Topology


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Boolean reflections for frames

The category Frm of frames is an algebraic [point-free) 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 point-sensitive 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 set-theoretic 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 Cantor-Bendixson 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

Harold Simmons (Manchester Univ., UK)

May 16, 2006




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