Homology, Homotopy and Applications

Volume 19 (2017)

Number 1

Homology of dendroidal sets

Pages: 111 – 134

DOI: http://dx.doi.org/10.4310/HHA.2017.v19.n1.a6

Authors

Matija Bašić (Faculty of Science, University of Zagreb, Croatia)

Thomas Nikolaus (Max Planck Institute for Mathematics, Bonn, Germany)

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 $\mathcal{K}(X)$ as discussed in previous work by the authors. Since these homology groups are sometimes computable we can identify some spectra $\mathcal{K}(X)$ which we could not identify before.

Keywords

homology, dendroidal set

2010 Mathematics Subject Classification

19D55, 55P42, 55U15, 55U35

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Published 6 June 2017