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
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