Strong pro-fibrations and ANR objects in pro-categories
S. Mardesic and T. B. Rushing called a system map \(\mathbf f: \mathbf X \to \mathbf Y \) a \textsl{pro-fibration} whenever it satisfies a homotopy lifting property (HLP) for all spaces \(Z\). A \textsl{strong pro-fibration} satisfies the stronger (SHLP) which differs from the (HLP) by the requirement that the required lifting \(\tilde H: Z \times I \to X_\lambda\) of a homotopy \(H: Z \times I \to Y_\mu\) for appropriate indexes \(\lambda, \mu\), has the property that if \(H\) is stationary on some \(\{z\}\times I,\;z \in Z\), so is \(\tilde H\). This describes precisely the difference between Hurewicz fibrations and regular Hurewicz fibrations in the category \textbf{Top} so that one should probably call the author's \textsl{strong} pro-fibrations \textsl{regular} pro-fibrations. The second new concept of the author is that of an ANR object: A system \(\{X_\lambda, p_{\lambda \lambda'}, \Lambda\}\) is called an ANR object if for each \(\lambda \in \Lambda\) there exists a \(\mu\geq \lambda\) so that \(p_{\lambda\mu}\) factors over an ANR. Every ANR system is an ANR object. The converse holds for ANR towers in \textbf{pro-Top}. The full subcategory of \textbf{pro-Top} of ANR objects is called \(\mathcal{ANR}\). The author tries to verify a very weak form of the axioms of a closed model category for \(\mathcal{ANR}\) with his strong pro-fibrations as fibrations and some notion of weak equivalences. It turns out that of these four properties two are satisfied, one is satisfied under a restriction while one is probably hopeless. Moreover in a last section the author deals with Postnikov decompositions of ANR towers. Furthermore the author tries to apply his theory which works only on \textbf{pro-Top} or some versions of \textbf{pro-Top} (like the category of towers) to shape theory which in his case always means ordinary shape, where fibrations are very problematic, because everything occurs already on the homotopy level. Although he mentions in his references the work of H. Thiemann who introduced a notion of a fibration for \textsl{strong} shape, he does not make any use of it.
scientific article; zbMATH DE number 5775738
scientific article