Homology of dendroidal sets

TL;DR

This paper introduces a homology theory for dendroidal sets via a chain complex, generalizing simplicial set homology, and proves its equivalence to the homology of associated spectra, enabling new spectrum identifications.

Contribution

It defines a chain complex for dendroidal sets, establishes a homology theory, and proves its isomorphism to spectrum homology, extending previous work.

Findings

Homology groups of dendroidal sets are computable.

Homology of dendroidal sets matches that of associated spectra.

New spectrum identifications are possible through this homology theory.

Abstract

We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups $H_n(X)$ as the homology groups of this chain complex. This generalizes the homology of simplicial sets. Our main result is that the homology of X is isomorphic to the homology of the associated spectrum K(X) as discussed in earlier work of the authors. Since these homology groups are sometimes computable we can identify some spectra K(X) which we could not identify before.