Dold–Kan correspondence for dendroidal abelian groups

Abstract

a b s t r a c tWe prove a Dold-Kan type correspondence between the category of planar dendroidal abelian groups and a suitably constructed category of planar dendroidal chain complexes. Our result naturally extends the classical Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes.

Homology of dendroidal sets

“…From our constructions it follows that Γ(C) underlies the dendroidal set Γ d (C) (which is in fact a dendroidal abelian group) and that Γ d (C) is fully Kan. This observation can be promoted to a dendroidal Dold-Kan correspondence (slightly different in spirit to the one in [GLW11] which only works for planar dendroidal sets). whereX → X is a normal, i.e.…”

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

“…The statement may be easily checked by the reader. It has been described in detail in Section 4 of [9] and Section 2.2.3 of [14].…”

Categorical Homotopy I. Quivers

“…Its maximal GReedy completion is clearly itself. This class of examples can be thought of as a generalization of [14] in four aspects. First, rooted trees is a subclass of sink-rooted quivers.…”

“…Second, we have more projections and sections in our setting. For example, for a source o, the maps S o and S r o are invisible in [14]. Third, we allow foldings.…”