Some obstacles to duality in topological algebra

Paul Bankston
1982 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Introduction. Functors $ \sé -» £ë, Y \3i ->seform an equivalence of categories (see [8,]) if T($(A)) ^ A and $(r(5)) ^ B naturally for all objects A ivoms/ and B from 3S. Lettings/* denote the opposite of s/ y we say that s/ and Se are dual if there is an equivalence between J/* and SB. Let r be a similarity type of finitary operation symbols. We let L T denote the first order language (with equality) using nonlogical symbols from r, and consider the class <Jé r of all algebras of type r as a
more » ... ras of type r as a category by declaring the morphisms to be all homomorphisms in the usual sense (i.e., those functions preserving the atomic sentences of L T ). If 3? is a class in^T (i.e., Jf C^T andj^ is closed under isomorphism), we view Jf as a full subcategory of^# T , and we define the order of S^ to be the number of symbols occurring in r. If 5f is a class of topological spaces and Stf is a class of algebras, let ¥ -Stif denote the category of "^-topological J^-algebras" (i.e., the topologies are in j^, the algebras are injT, and the operations are jointly continuous) plus continuous homomorphisms. A dual pair, for our purposes, is simply a pair (5S -Stf, $~ -<££) where the categories are dual to one another. (If 5 denotes the category of sets, we treat S-Siïf and^-S as identical with Sf and 3T respectively.) Beautiful examples of dual pairs in topological algebra are well known (see (0.1) below), and it is our intention in this paper to underscore the special nature of some of these examples by laying down fairly general conditions on the classes iS^,^", J^, ando£f ensuring the nonexistence of a duality between ¥'-jf and^-if. In the following, certain categories of special interest will be given abbreviations. (i) LCH = {locally compact Hausdorff spaces}, CH = {compact Hausdorff spaces}, and CCH, ZDCH, and EDCH denote respectively the connected, O-dimensional, and extremally disconnected objects inCH. (ii) AG = {abelian groups}, and TAG and TFAG denote respectively the torsion and torsion free objects in AG. (iii) SL = {semilattices}.
doi:10.4153/cjm-1982-008-6 fatcat:374vtxoq4zb4dofxhtel2jxbgu