Partition complexes and trees

TL;DR

This paper constructs a homotopy initial functor linking partition complexes to labeled trees, leading to new insights into operad bar constructions and providing a simplified proof of an existing equivalence.

Contribution

It introduces a homotopy initial functor from partition complexes to labeled trees, establishing an equivalence between bar constructions of operads and simplifying prior proofs.

Findings

Establishes a homotopy initial functor between partition complexes and trees.

Provides an equivalence between different operad bar constructions.

Offers an elementary proof of a known equivalence in the differential graded setting.

Abstract

We construct a homotopy initial functor from the partition complex of a finite set $A$ to a category of trees with leaves labelled by $A$. As an application, this provides an equivalence between different bar constructions of an operad. In the differential graded case, this gives a very elementary proof of an equivalence originally due to Fresse.