Batanin's category of pruned trees is Koszul

TL;DR

This paper proves that Batanin's category of pruned trees is Koszul, providing a minimal model and enabling new computations of homological invariants in related categorical structures.

Contribution

It establishes the Koszul property of Batanin's pruned trees category, facilitating minimal models and homological calculations.

Findings

Category of pruned trees is Koszul

Provides minimal differential graded model

Enables computation of Tor and Ext functors

Abstract

The category of pruned trees has been defined by M. Batanin with the aim of understanding the cell structure of certain E_n-operads in categorical terms. The objects of this category are planar trees with n levels so that all leaves are at the top level of the tree. The goal of this article is to prove that the category of pruned trees is Koszul. This result gives us a minimal differential graded model of this category, small complexes to compute Tor and Ext functors in associated categories of diagrams, and allows us to generalize a recent result of M. Livernet and B. Richter about the interpretation of E_n-homology in terms of categorical Tor functors.