Dold-Kan correspondence for dendroidal Abelian groups (Q531375)

The authors generalize the equivalence of categories between simplicial abelian groups and non-negatively graded chain complexes (the Dold-Kan correspondence) to an equivalence of categories between planar dendroidal abelian groups and a suitably constructed category of planar dendroidal chain complexes. The role played by the simplicial category \(\Delta\) in the Dold-Kan correspondence is played in this new correspondence by the category of planar rooted trees. A rooted tree is a connected finite graph with no loops equipped with a distinguished vertex called the output vertex. A planar rooted tree is a rooted tree such that each vertex is equipped with a linear ordering on the set of input edges. Rooted trees are used for studying the homotopy theory of coloured operads and their algebras. In the context of non-symmetric coloured operads, the role of dendroidal sets is played by planar dendroidal sets which are presheaves over planar rooted trees.