Constructions of En operads

Throughout this talk I will use the following conventions and notations. I will primarily consider operads in the category of compactly generated Hausdorff topological spaces having the homotopy type of CW-complexes. When I refer to simplicial operads or operads in the category of posets, it will be understood that they can be converted to topological operads by taking geometric realization or geometric realization of the nerve, respectively. I will only consider operads A satisfying A(0) = *. This allows us to define degeneracy maps by composing with * ∈ A(0). We will refer to collections of spaces A = {A(k)} k≥0 equipped with symmetric group actions and such degeneracy maps, satisfying the evident relations, but without any further operad composition maps, as preoperads. (We refer the reader to Berger's talk in this volume [2] (or [3]) for a precise definition.)