Morphism complexes of sets with relations

TL;DR

This paper studies the homotopy theory of $r$-sets, generalizing graph Hom complexes to hypergraphs, introducing singular complexes, and establishing their homotopy equivalence and folding theorems.

Contribution

It introduces singular complexes for $r$-sets, proves their homotopy equivalence with Hom complexes, and generalizes $ imes$-homotopy theory to hypergraphs with a folding theorem.

Findings

Singular complexes are homotopy equivalent to Hom complexes.

Established a folding theorem for hypergraphs.

Generalized $ imes$-homotopy theory to $r$-sets.

Abstract

Let $r$ be a positive integer. An $r$-set is a pair $X= (V(X),R(X))$ consisting of a set $V(X)$ with a subset $R(X)$ of the direct product $V(X)^r$. The object of this paper is to investigate the Hom complexes of $r$-sets, which were introduced for graphs in the context of the graph coloring problem. In the first part, we introduce simplicial sets which we call singular complexes, and show that singular complexes and Hom complexes are naturally homotopy equivalent. The second part is devoted to the generalization of $\times$-homotopy theory established by Dochtermann. We show the folding theorem for hypergraphs which was partly proved by Iriye and Kishimoto.