18 ANTHONY V. PHILLIPS and DAVID A. STONE

Lemma 1.14 Let K\, K* be coalgebras with comultiplications V1,

V2,

and counits

e1, e2

respectively. Let T: K\ ® I1 — • Kl® I1 be de-

fined by T(CX®C2) = (-ly^C^C1), where r = dim C\s = dim C2.

Then / ^ ®

if2

is a coalgebra with respect to the comultiplication given

by the composition

Kl ® I1

V

^

2 (/C1

0 Kl) ® {Kl ® Kl)

x*l?x

(Kl ®

tf2)

® (Kl ®

A'2).

The counit on I1 ®

if2

is

e1

®

e2.D

1.15 For example, let A be a simplicial complex and K a cubical one,

both locally ordered. Then on A* ® K+ we have the comultiplication V

given by the following formula, in which dim a = fc, dim C = n, and II

is a two-fold partition of { 1 , . . . , n) with [II'I = p, |II"| = n — p.

V(a ® C) = J2 E e(n)(-l) f c -^(a

b

(i ) ® *h(C)) 8 (r«(j) ® 5

n

(C)).

i=o n