Vladislav Zemlyanoy - On representing fine shape of all metrizable spaces
Monday, July 24, 1pm GMT
Speaker: Vladislav Zemlyanoy
Title: On representing fine shape of all metrizable spaces
Abstract: The strong shape category of compact metrizable spaces is known to have multiple equivalent definitions, and many properties and results are known for it, such as the invariance of Steenrod-Sitnikov homology and Čech cohomology. Fine shape, as defined by Melikhov, is an extension of strong shape to noncompact metrizable spaces that keeps invariance of both; its definition is also far simpler than that of most noncompact shape theories. This raises the question of extending other known results for compact strong shape to (noncompact) fine shape. In that vein, representing fine shape through approaching maps of absolute retracts, we can show it to be a specific left fraction category, thus extending Cathey's definition of compact strong shape. In the process, we raise two more questions yet to be answered fully: one about noncompact version of Mrozik's mapping cylinder of an approaching map, and one about finding a model structure for which fine shape is the homotopy category. The latter in particular could hardly be done for compact spaces only, as the arising path spaces would in all probability be noncompact.