Simple homotopy theory and nerve theorem for categories
The paper develops some concepts of simple homotopy theory in the context of combinatorial topology. The discussion concentrates in particular on \(\Delta\)-complexes and loop-free categories. The former is a general form of a semi-simplicial set, while the latter is a category with trivial endomorphism sets, and where the existence of a morphism \(x\to y\) excludes the existence of a morphism the other way. Both concepts are well used, specifically in topological data analysis. The paper extends work of \textit{J. A. Barmak} and \textit{E. G. Minian} [Adv. Math. 218, No. 1, 87--104 (2008; Zbl 1146.57034)]. In particular a version of Quillen's theorem A. The paper also gives two generalisations of the classical nerve theorem. The first is to what is referred to as quasi-good covers, i.e. open covers such that every non-empty finite intersection has contractible connected components. The second generalisation is to \(\pi_1\)-good covers, i.e. covers such that every non-empty finite intersection is a \(K(\pi,1)\) space. The paper also offers an application of the nerve theorem to topological data analysis.
scientific article; zbMATH DE number 7318664
scientific article