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• W2890393022 abstract "Cech 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 $K_alphasubset X$. 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 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 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 $|P|$." @default.
• W2890393022 created "2018-09-27" @default.
• W2890393022 creator A5036474311 @default.
• W2890393022 date "2018-08-30" @default.
• W2890393022 modified "2023-09-27" @default.
• W2890393022 title "Lim colim versus colim lim. II: Derived limits over a pospace" @default.
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• W2890393022 hasPublicationYear "2018" @default.
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