Configuration spaces form a Segal semi-dendroidal space

TL;DR

This paper demonstrates how semi-dendroidal spaces satisfying the Segal condition can model configuration spaces of points in an n-dimensional ball, highlighting their utility in describing 'up-to-homotopy' operads.

Contribution

It introduces a semi-dendroidal space satisfying the Segal condition that models configuration spaces, providing a new tool for understanding operads up to homotopy.

Findings

Semi-dendroidal space satisfies the Segal condition.

Evaluation at a k-corolla yields configuration space of k points.

Illustrates utility of dendroidal objects in homotopy operad theory.

Abstract

The purpose of this short note is to illustrate the utility of (semi-) dendroidal objects in describing certain 'up-to-homotopy' operads. Specifically, we exhibit a semi-dendroidal space satisfying the Segal condition, whose evaluation at a k-corolla is the space of ordered configurations of k points in the n-dimensional unit ball.