Limit of colimits versus colimit of limits II

This talk essentially does not depend on the previous one. Last week we encountered a situation where limits turn out to commute with colimits, namely the general additivity axiom. (This axiom, along with the map excision axiom and the Eilenberg–Steenrod axioms, suffices to uniquely characterize Steenrod–Sitnikov homology and Čech cohomology of Polish spaces.)
This time we will study a situation where limits do not commute with colimits (and we will discuss specific examples of non-commutativity), but their “commutator” can be computed in terms of lim$^1$ (which will be defined and explained) and a new functor lim$^1_{fg}$. Namely, there are two well-known approximations to the Steenrod–Sitnikov homology $H_n(X)$ of a Polish space $X$: “Čech homology” $qH_n(X)$, which is a limit of colimits of homology groups of finite simplicial complexes, and “Čech homology with compact supports” $pH_n(X)$, which is a colimit of limits of homology groups of the same finite simplicial complexes.
The natural map $pH_n(X)\to qH_n(X)$ is generally neither injective (P. S. Alexandrov, 1947) nor surjective (E. F. Mishchenko, 1953), but it remains an open problem whether it is surjective when $X$ is locally compact. We show that for a locally compact $X$ the dual map in cohomology $pH^n(X)\to qH^n(X)$ is surjective and compute its kernel. This computation has applications to embeddability of compacta in Euclidean spaces. In homology, we show that $pH_n(X)\to qH_n(X)$ is surjective and compute its kernel when $X$ is a “compactohedron”, i.e. contains a compactum whose complement is homeomorphic to a simplicial complex.