Morphism complexes of sets with relations (Q5964057)

Let \(r>0\) be a positive integer. A pair \(X:=(V,R)\) with \(V\) a set and \(R\) the \(r\)-fold product of \(V\) is an \(r\)-set. From this definition, the Hom complex \(\mathrm{Hom}(X,Y)\) is defined as the poset of multi-homomorphisms from \(X\) to \(Y\). One of the main results of this paper is to relate this object to the singular complex, a categorical type construction also based on \(r\)-sets \(X\) and \(Y\). It is shown that there is a natural homomorphism between \(|\mathrm{sing}(X,Y)|\) and \(|\mathrm{Hom}(X,Y)|\). Furthermore, the authors investigate the strong homotopy type of posets and finite simplicial complexes of Barmak and Minian as a natural generalization of the \(x\)-homotopy theory of Dochtermann. The authors prove equivalent notions of a strong homotopy equivalence in terms of homotopy equivalences of associated poset maps. This leads to a new proof of what the authors call the folding theorem; that is, for \(r\)-sets \(X\) and \(Y\) along with a vertex \(x\in X\) with certain properties, the induced maps \(i^*: \mathrm{Hom}(X,Y)\to \mathrm{Hom}(X\backslash x,Y)\) and \(i_*: \mathrm{Hom}(Y,X\backslash x)\to \mathrm{Hom}(Y,X)\) on the inclusion \(i: (X\backslash x)\to X\) are strong equivalences.