Classifying spaces of categories and term rewriting (Q2762219)

\textit{K. S. Brown} and \textit{R. Geoghegan} [Invent. Math. 77, 367-381 (1984; Zbl 0557.55009)] introduced the notion of a collapsing scheme for a simplicial set. This specified classes \(E\), \(R\), and \(C\) of non-degenerate simplices are called Essential, Redundant and Collapsible, respectively. These were to satisfy certain axioms. The main result of their theory was that, if one has a collapsing scheme on \(X\), say, the geometric realisation of \(X\) has the homotopy type of a CW-complex \(\mathcal{E}(X)\), whose \(n\)-cells correspond to its essential \(n\)-simplices. NEWLINENEWLINENEWLINEThe classifying space of a category \({\mathcal C}\) is obtained by geometric realisation of its nerve, however by reason of its construction it may be difficult to study it as there will be a large number of cells in all dimensions. In this paper a category of simplicial sets with collapsing scheme is introduced and the Brown-Geoghegan construction, \(\mathcal{E}\), is shown to be functorial, and the canonical homotopy equivalence, \(|X|\to \mathcal{E}(X)\), to be a natural transformation. \textit{K. S. Brown} [in: Algorithms and classification in combinatorial group theory, Publ. Math. Sci. Res. Inst. 23, 137-163 (1992; Zbl 0764.20016)] showed how to derive a collapsing scheme from a rewrite system or a monoid presentation. NEWLINENEWLINENEWLINEHere the author extends this to presentations of categories and studies the nerve of the category \(\mathcal{IR}\) of normal forms of a complete rewrite system \(\mathcal{R}\). If \(R\) is a complete presentation of a small catgory \({\mathcal C}\), then the collapsed space \(\mathcal{E}(N\mathcal{IR})\) is (up to homotopy) the classifying space \(B{\mathcal C}\) of \({\mathcal C}\). Several useful examples are given as illustrations.