The Assembly Tower and Some Categorical and Algebraic Aspects of Frame Theory

Abstract : This thesis studies the framework arising in the algebraic and categorical description of general (or point-set) topology. Classically, a topological space is a set with structure, the structure being its collection of open sets, which taken together determine an abstract notion of proximity. The collection of all such open sets forms a special kind of complete lattice, and it is a class of complete lattices (frames) motivated by these examples that is the focus of algebraic study-in short, one dispenses with the points and studies the algebra of open sets. This method has had successes not only in general topology, but has also found application in such diverse areas as logic, topos theory, and even computer science. It is not these specific areas of application, however, with which the thesis is primarily concerned; rather, it is that part of the theory which they all share: the category of frames. This category has as a sub-category the category of complete Boolean algebras, and these two categories stand in much the same relation as do the categories of topological spaces and sets. As with sets and spaces, complete Boolean algebras are in some ways better behaved categorically than frames, and so the former provides a potential source of information about the latter. For the purpose of obtaining this information, a construction for frames, called the assembly tower and present already at the beginnings of the subject, is studied systematically and in this way found to be a key tool for uncovering both structural and algebraic properties of frames.