Strongly fibered objects and spaces (Q1688202)

Let \(\mathcal{C}\) be a groupoid enriched (g.e.) category with a model category structure having a functorial cylinder, then it is known that the category, \(Pro(\mathcal{C})\), of inverse systems in \(\mathcal{C}\) also has a g.e. model category structure. This allows discussions of strong shape theory within the context of \(\mathcal{C}\) and its model category, relative to a full subcategory, \(\mathcal{K}\), all of whose objects are fibrant; see [\textit{L. Stramaccia}, J. Homotopy Relat. Struct. 12, No. 2, 433--446 (2017; Zbl 1372.55015)]. (The classical case was \(\mathcal{C}=\mathsf{TOP}\) and \(\mathcal{K}=\mathsf{ANR}\).) Generalising a classical topological notion, the author introduces the notion of a strongly fibered object, corresponding to limits of fibrant inverse systems in \(Pro(\mathcal{C})\), and denotes by \(\mathcal{F(K)}\) the full subcategory of \(\mathcal{C}\) determined by these objects. The main theorem shows that the shape theory of the pair \((\mathcal{C},\mathcal{F(K)})\) realises the strong shape category of \((\mathcal{C},\mathcal{K})\), thus generalising the classical result in the topological context.