×

The use of semisimplicial complexes in strong shape theory. (English) Zbl 0781.55006

Amongst the known descriptions of strong shape theory, one due to F. W. Bauer [The advantages of strong shape theory, in “Topology, general and algebraic topology, and applications”, Proc. Int. Conf., Leningrad 1982, Lect. Notes Math. 1060, 119-128 (1984; Zbl 0553.55007)] uses the language of \(\infty\)-categories. The author here puts forward an approach based on simplicial enriched categories, and homotopy coherence using various ideas from the work of R. M. Vogt [Math. Z. 134, 11-52 (1973; Zbl 0276.55006)] and J.-M. Cordier and the reviewer [Math. Proc. Camb. Philos. Soc. 100, 65-90 (1986; Zbl 0603.55017)]. Using large simplicial sets, he constructs for each space a simplicial analogue of a comma category of ‘nice spaces’ under \(X\). He then uses these to define the strong shape theory category and a strong shape category of pairs.
Reviewer: T.Porter (Bangor)

MSC:

55P55 Shape theory
18D20 Enriched categories (over closed or monoidal categories)
54C56 Shape theory in general topology
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
55U10 Simplicial sets and complexes in algebraic topology
55U35 Abstract and axiomatic homotopy theory in algebraic topology
PDFBibTeX XMLCite