An interpretation of E_n-homology as functor homology

TL;DR

This paper demonstrates that E_n-homology of non-unital commutative algebras can be understood through functor homology on a category of planar trees, linking various homology theories like Hochschild and Gamma homology.

Contribution

It provides a new categorical interpretation of E_n-homology as functor homology, connecting multiple homology theories via natural maps.

Findings

E_n-homology described as functor homology on planar trees

Connections established between Hochschild, higher order, and Gamma homology

Natural maps relate different E_n-homology theories

Abstract

We prove that E_n-homology of non-unital commutative algebras can be described as functor homology when one considers functors from a certain category of planar trees with n levels. For different n these homology theories are connected by natural maps, ranging from Hochschild homology and its higher order versions to Gamma homology.