An operad (O, â—¦i) consists of a collection {O(n)}n≥1 of objects and maps â—¦i : O(m) × O(n) → O(m + n − 1) for m, n ≥ 1 satisfying the relations manifest in the example EndX . May’s original definition corresponds to simultaneous insertions into all possible positions of inputs into f ∈ Map(Xn, X). In most examples, the structures are “manifest” without appeal to the technical definitions; as Frank Adams used to say, to operate the machine, it is not necessary to raise the bonnet (look under the hood). It helps to see graphic examples of operads. Two kinds that are particularly important are the tree operads and the little cubes (or disks) operads. Let T (n) be the set of (nonplanar) trees with 1 root and n leaves labeled (arbitrarily) 1 through n. The collection T = {T (n)}n≥1 of sets of trees forms an operad by grafting the root of g to the leaf of f labeled i, as in Figure 1. The little n-cubes operad Cn = {Cn(j)}j≥1, where Cn(j) consists of an ordered collection of j n-cubes linearly embedded in the standard n-dimensional unit cube In with disjoint interiors and axes parallel to those of In.