Characterizing shape theories by Kan extensions
The author considers the study of the pairs (\textbf{C,K}) of categories, where \({\mathbf K}\) is a proreflective (or dense) subcategory of \({\mathbf C}\), which means that every object of \({\mathbf C}\) admits a \({\mathbf K}\)-expansion. Such a pair (\textbf{C,K}) is a germ for a shape theory and it is called a pre-shape theory. Let \textbf{Cat} denote the metacategory of all categories and let \textbf{Cat}* be its subcategories of all categories having inverse limits. Let (\textbf{C,K}) be a fixed pre-shape theory and let \(F:{\mathbf K}\to {\mathbf A}\), \({\mathbf A}\in \)\textbf{Cat}*, be a given functor. This functor defines in a natural way a functor \(\check F:{\mathbf C}\to {\mathbf A}\) which is called a Čech extension of \(F\). The author proves that this extension coincides, up to natural isomorphisms, with the pointwise right Kan extension Ran \(F:{\mathbf C}\to{\mathbf A}\) of \(F\) along the embedding \(E:{\mathbf K}\subset {\mathbf C}\). Moreover, Ran \(F\) is preserved by every functor \(G:{\mathbf A}\to {\mathbf B}\) which commutes with inverse limits. (For the notion of right Kan extension see [\textit{S. MacLane}, Categories for the working mathematician, Grad. Texts Math. 5 (1971; Zbl 0232.18001), p. 232]). With these preliminaries, the author proves the following main results: Theorem 3.1. Let (\textbf{C,K}) be a pair of categories. Assume that Ran \(e_{\mathbf K}\) exists and is preserved by Pro \(E\). Then (\textbf{C,K}) is a pre-shape theory with Ran \(e_{\mathbf K}\) as its proreflector. (Here \(e_{\mathbf K}: {\mathbf K}\to\text{Pro} {\mathbf K}\) is the natural embedding). Theorem 3.2. A pair (\textbf{C,K}) is a pre-shape theory if and only if it has the Kan extension property, that is: for every functor \(F:{\mathbf K}\to {\mathbf A}\), \({\mathbf A}\in \)\textbf{Cat}*, Ran \(F\) exists and it is preserved by every functor \(G:{\mathbf A}\to {\mathbf B}\) which commutes with inverse limits. As examples of pre-shape theories the following pairs of homotopy categories are obtained: \(\bullet\) (\textbf{HTop, HlfPol}), topological spaces, locally finite polyhedra. \(\bullet\) (\textbf{HCompT}\(_2,\) \textbf{HfPol}), compact Hausdorff spaces, finite polyhedra. \(\bullet\) (\textbf{HSMet, HclfPol}), separable metric spaces, countable locally finite polyhedra. \(\bullet\) (\textbf{Hfd Norm, HfdPol}), finite dimensional normal spaces, finite dimensional polyhedra. \(\bullet\) (\textbf{HPar, HfdPol}), finite dimensional Hausdorff paracompact spaces, finite dimensional polyhedra. Finally, a well-known result from topological categories is extended in the general case: Theorem 3.4. A pair of categories (\textbf{C,K}) is a pre-shape theory whenever \(K\) is closed with respect to limits (possibly large) and Ran \(e_{\mathbf K}\) exists pointwise.
scientific article; zbMATH DE number 1787503
scientific article