On the compact varinjlim-functor in the category of compact groups

If G is a discrete abelian group and K its Pontryagin dual then, for widetilde X a finite complex or compact polyhedron, the homology and cohomology groups are also linked via duality H_n(X;G) with H^n(X;K), etc. In 1959 Aleksandrov asked if the dual groups of homology or cohomology groups of an arbitrary topological space X with coefficients in G have an independent geometric meaning. He listed Čech homology and cohomology, Sitnikov homology, Čech homology and Čech cohomology with compact supports and pointed out that there were satisfactory solutions known for the first and last cases. These solutions used a compact direct limit of a direct system of compact groups due to G. S. Čogošvili [Mat. Sbornik N.S. {bf 28(70)} (1951), 89--118; [msn] MR0041431 (12,846f) [/msn]]. In this paper a description of all dual groups to these homology and cohomology groups is given. This is given using constructions of a compact homotopy limit of chain complexes and a resulting strong compact (co)homology group of a direct system of compact Hausdorff (co)chain complexes. The proofs also introduce a "scalar product" of inverse and direct limits which may be of independent interest.