Remark on the Simplicial-cosimplicial

We show that the existence of canonical representatives for the elements of the tensor product (coend) of a simplicial and a cosimplicial set depends only on the Eilenberg-Zilber property of the given cosimplicial set. Thus the second condition which is used in [5] for achieving this result is superfluous. Let X: z"P S be a simplicial set and Y: A -S a cosimplicial set. We consider X as a N-graded set with A acting on the right and correspondingly Y is a N-graded set with A acting on the left. The Eilenberg-Zilber Lemma states that every x E X has a unique decomposition (1) x = x+xO with x+ nondegenerate and xo surjective. We assume that Y has the dual property, i.e. every y E Y has a inique decomposition (2) y = y+yO with y+ injective and y0 interior. (That the proof of the Eilenberg-Zilber Lemma fails to be dualizable depends on the fact, that any surjective map in uniquely determined by the set of its sections; but different injective maps with the same one-element domain and the same codomain have the same set of retractions. Thus the Eilenberg-Zilber property for cosimplicial sets is a real restriction; see 12,4.4 and 41 for a further discussion of this phenomenon.) Now take (3) T, = ((x, y) I x e X, y e Y, degree x degreey = n)