Configuration space of intervals with partially summable labels

TL;DR

This paper constructs a generalized configuration space of intervals with partially summable labels, extending previous models, and proves its homotopy equivalence to a loop space under certain conditions.

Contribution

It introduces a new configuration space model with partially summable labels and establishes its homotopy equivalence to a loop space of a classifying space.

Findings

The new configuration space extends previous models.

An approximation theorem is generalized to this setting.

The space is shown to be homotopy equivalent to a loop space under certain conditions.

Abstract

A configuration space of intervals in $\mathbb R^1$ with partially summable labels is constructed. It is a kind of an extension of the configuration space with partially summable labels constructed by the second author and at the same time a generalization of the configuration space of intervals with labels in a based space constructed by the first author. An approximation theorem of the preceding configuration space is generalized to our case. When partially summable labels are given by a partial abelian monoid $M,$ we prove that it is weakly homotopy equivalent to the space of based loops on the classifying space of $M$ under some assumptions on $M$.