Universal properties of variations of the little cubes operads

Given a map B → B Top(n) of spaces, one can define a version EB of the little cubes operad, whose construction is due to Lurie. We show that EB enjoys the universal property that, for every ∞-operad O, an operad map EB → O is equivalent to a Top(n)- equivariant map B ×B Top(n) E Top(n) → Map(En,O). This gives us an explicit diagram exhibiting EB as a colimit of En parametrized by B. It also shows that locally constant factorization algebras satisfy descent, reproving a recent theorem of Matsuoka.