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The Dold-Kan Correspondence is an equivalence between the category of differential graded objects and the category of simplicial objects on an abelian category. This equivalence is best understood within the context of Quillen model categories. However, a more straightforward interpretation using the representation theory of small categories is possible. We will demonstrate that the Dold-Kan equivalence can be expressed through specific induction and restriction functors, paving the way for similar equivalences for crossed-simplicial objects. There are extensions to the Dold-Kan Correspondence in this context, with the Dwyer-Kan equivalence between the category of duplicial objects and the category of cyclic objects over an abelian category being a notable example. We will also show that the Dwyer-Kan equivalence can be incorporated into the framework we initially developed for the Dold-Kan Correspondence. Lastly, we will discuss further extensions.

This research is a joint work with my PhD student, Haydar Can Kaya.