“Purity and pure-injectivity for topological modules.” In Models, Logics, and Higher-Dimensional Categories: A Tribute to the Work of Mihàly Makkai, CRM Proceedings and Lecture Notes 53, eds. Bradd Hart, Thomas G. Kucera, Anand Pillay, and Robert A. G. Seely (Providence: American Mathematical Society: 2011). 55-78. (Co-written with Thomas G. Kucera.)

Dedicated to Professor Mihàly Makkai on the occasion of his seventieth birthday.

Abstract

A logic suitable for topological structures was introduced by McKee and Sgro in the 1970s and developed further by Garavaglia, Ziegler and Kucera. This logic satisfies the compactness theorem, a Löwenheim–Skolem theorem, and a Löwenheim theorem.

The model theory of modules is very rich with strong connections to standard questions of algebra, primarily based on the partial elimination of quantifiers down to positive primitive formulas. Theories of topological modules also have a partial elimination of quantifiers, down to “topological positive primitive formulas,” so there is some hope to see if a development of these theories can be made that parallels the development of the model theory of ordinary modules.

In his M.Sc. thesis, Enns investigated whether there are good definitions of “pure embedding” and of “pure-infective module” for topological modules in this context. The answer is “mostly not.” We report on these results and related results from the literature here.

Summary

We have considered the possibility of developing as useful model theory of topological modules (useful for the purposes of topological algebra) in the languages ℒ_t or ℒ_m.

• Any approach to this problem that involves the use of open-set parameters, or imposes topological constraints on the morphisms between structures, seems unlikely to produce anything useful.

• If we consider the category of topological modules with module homomorphisms that preserve topological positive primitive formulas, we can get some results of a model-theoretic nature, but appear to have little of topological interest.

• Nothing we have discussed here places any limits on what might be accomplished by other means. In particular, it places no restrictions on what might be accomplished within the (non-axiomatizable) class of locally compact abelian groups or modules.