Simplicial Coendomorphism Operads and Coalgebras

In a recent work of Moreno-Fernandez, Wierstra and the present author [‘A recognition principle for iterated suspensions as coalgebras over the little cubes operad’, Preprint, 2022, arXiv:2210.00839], a coendomorphism operad in the category of pointed topological spaces endowed with the wedge sum was introduced. In this paper, we construct an analogue completely internal to the category of simplicial sets with the goal of defining simplicial coalgebras. As an application, we show that simplicial n -fold suspensions are coalgebras up to coherent homotopy over the Barratt–Eccles Eₙ -operad provided they have finitely many nondegenerate simplices.