Lim colim versus colim lim. II: Derived limits over a pospace

TL;DR

This paper explores how to reconstruct Čech cohomology and Steenrod-Sitnikov homology of a space from the homology of its compact subsets or neighborhoods using a spectral sequence with corrected derived limits that account for topological structures.

Contribution

It introduces a method to reconstruct homology and cohomology from inverse systems by correcting derived limits with a natural topology on the indexing set, extending classical approaches.

Findings

Corrected derived limits match classical ones in discrete topology cases.

A metrizable topology on the order complex is constructed, consistent across multiple approaches.

The method applies to spaces with natural topologies, like hyperspaces with the Hausdorff metric.

Abstract

\v{C}ech cohomology $H^n(X)$ of a separable metrizable space $X$ is defined in terms of cohomology of its nerves (or ANR neighborhoods) $P_\beta$ whereas Steenrod-Sitnikov homology $H_n(X)$ is defined in terms of homology of compact subsets $K_\alpha\subset X$. We show that one can also go vice versa: in a sense, $H^n(X)$ can be reconstructed from $H^n(K_\alpha)$, and if $X$ is finite dimensional, $H_n(X)$ can be reconstructed from $H_n(P_\beta)$. The reconstruction is via a Bousfield-Kan/Araki-Yoshimura type spectral sequence, except that the derived limits have to be "corrected" so as to take into account a natural topology on the indexing set. The corrected derived limits coincide with the usual ones when the topology is discrete, and in general are applied not to an inverse system but to a "partially ordered sheaf". The "correction" of the derived limit functors in turn involves constructing a "correct" (metrizable) topology on the order complex $|P|$ of a partially ordered metrizable space $P$ (such as the hyperspace $K(X)$ of nonempty compact subsets of $X$ with the Hausdorff metric). It turns out that three natural approaches (by using the space of measurable functions, the space of probability measures, or the usual embedding $K(X)\to C(X;\mathbb R)$) all lead to the same topology on $|P|$.