Monoid Structures on Indexed Containers

Containers represent a wide class of type constructions relevant for functional programming and (co)inductive reasoning. Indexed containers generalize this notion to better fit the scope of dependently typed programming. When interpreting types to be sets, a container describes an endofunctor on the category of sets while an I-indexed container describes an endofunctor on the category Set^I of I-indexed families of sets. We consider the monoidal structure on the category of I-indexed containers whose tensor product of containers describes the composition of the respective induced endofunctors. We then give a combinatorial characterization of monoids in this monoidal category, and we show how these monoids correspond precisely to monads on the induced endofunctors on Set^I. Lastly, we conclude by presenting some examples of monads on Set^I that fall under our characterization, including the product of two monads, indexed variants of the state and the writer monads and an example of a free monad. The technical results of this work are accompanied by a formalization in the proof assistant Cubical Agda.