Homology of dendroidal sets

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 [BN14] and [Nik14]. Since these homology groups are sometimes computable we can identify some spectra K(X) which we could not identify before.

Augmented Homotopical Algebraic Geometry

“…However, what does hold is the fact that Ho( A Kan ) is a simplicial category. Using the tools presented in [5,Proposition 5.4.4] and [35,Lemma 3.8.1] we can formally prove the following, although the result is non-surprising due to the way we have constructed the classes of maps in our model structure. Remark 4.30.…”

“…Remark 4.1. The material presented in this section draws heavily on the construction of the stable model structure for dendroidal sets [5], which, in turn, follows the presentation of [45] and [35]. In fact, one sees that the definition of the augmentation category is rigid enough that the arguments relating to the model structure developed in [5] are simply altered in a consistent manner, replacing instances of Ω with A.…”

“…This section will lead to the choice of the quantifier stable to describe Ω-augmentation. The results in this section are built on [5], where the model Ω Kan was explicitly described. Note that in the aforementioned reference, the Kan model structure is called the stable model structure.…”

On the combinatorics of faces of trees and anodyne extensions of dendroidal sets

“…Also, the case where both maps are in the category of open dendroidal sets shows that the corresponding model structure on open dendroidal sets is compatible with the colax monoidal structure. This paper is based on one chapter of the author's PhD thesis ( [1]). The material has been significantly rewritten in order to simplify the presentation and make the combinatorial conditions more natural, but this has not changed the underlying content and the main results as stated in the chapter of the thesis.…”

“…This paper is based on one chapter of the author's PhD thesis ( [1]). The material has been significantly rewritten in order to simplify the presentation and make the combinatorial conditions more natural, but this has not changed the underlying content and the main results as stated in the chapter of the thesis.…”