Strongly fibered objects and spaces. (English) Zbl 1405.55022
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 [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.
Reviewer: Timothy Porter (Llandegfan)
MSC:
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
| 55P55 | Shape theory |
| 18D20 | Enriched categories (over closed or monoidal categories) |
Keywords:
inverse system; groupoid enriched category; pseudo-natural transformation; homotopy limit; shape category; strong shape categoryCitations:
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