turkmath.org

The Dold-Kan Theorem is a statement about the equivalence between the category of simplicial objects and the category of chain complexes over an abelian category, after certain natural localizations. This equivalence is a Quillen model categoric equivalence, with different restricted versions of it appearing in literature. In this talk, we will explore an extension to the Dold-Kan equivalence by replacing the simplicial category with a crossed simplicial group. Crossed simplicial groups are obtained from the simplicial category through a bicrossed product between the simplicial category and certain collections of (finite) groups. Connes' cyclic category being an example of this construction. While previous attempts to extend the Dold-Kan to the Connes' cyclic category and beyond involved constructing artificial categories in place of the category of dg-objects over an abelian category, we will pursue a new approach in this talk. Specifically, we will show that the Dold-Kan result can be reformulated using certain induction and restriction functors, providing a clear path for a natural extension of the Dold-Kan to crossed simplicial groups when the groups used are all finite. This is an ongoing joint research project with my PhD student, Haydar Can Kaya.