A model for the group completions of configuration spaces of points via configurations of labeled intervals/strings (cf. at group-completed configuration spaces of points):
Shingo Okuyama: The Space of Intervals and an Approximation to (2003) [pdf, pdf]
Shingo Okuyama: A simple solution for a group completion problem, Trends in Mathematics 7 1 (2004) 69-74 [pdf, pdf]
Shingo Okuyama: The space of intervals in a Euclidean space, Algebr. Geom. Topol. 5 (2005) 1555-1572 [arXiv:math/0511645, doi:10.2140/agt.2005.5.1555]
Shingo Okuyama, Kazuhisa Shimakawa, Interactions of strings and equivariant homology theories, Geom. Topol. Monogr. 10 (2007) 333-346 [arXiv:0903.4667, doi:10.2140/gtm.2007.10.333]
Shingo Okuyama: Configuration space of intervals with partially summable labels, talk at Shinshu University (2018) [pdf, pdf]
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