Simple homotopy theory and nerve theorem for categories (Q1998811)

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.