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By Stanislaw Saks

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I(0i)k' 2. t is equal to the smallest value of k such that i + 1 — £ " = 1 (Pi )i> < ]C»=i (A )('• 3. ,-);/ denotes the i' term of the sequence &. , conjugated by the transposition that interchanges 1 and i. We now pass AJOZ through the other terms to get ftt^l(n) 0l . . ( - 1 ) ' £ ' « " = > A,- 0i, tl"«l(r t ) ot .. ,#,-•• ok A t ) , & } o (-i)*'+l«li+*'i /»T l ^ l| ('-i) °i .. ( - i y E 1 ; > « A , - o,-, ]\p'\rt) ot . ^«\rk) ok A t ) But, except for a factor of (—l) fc_1 in the definition of tk, this is the same as what we would have gotten by evaluating (—l)-7 l0

H o m z ( ® } = 1 9l[C]^, Cv) sends tensor products of maps to maps of the tensor products of their domains and ranges. 15. 2. ((g)S^C], ,C") i=i (g)Homzs, . ( * rn — ck ITERATING THE COBAR CONSTRUCTION 47 of degree k — 2: jik = ( - i ) * l « l P o 0 | m o jW z { A , . . fcl :E-l/3il9l[C^i 0 . . 0 E-'^ljRtC]^ where a — Yli=\ Pi anc ^ m e summation of sequences is taken elementwise. Here the {(3i} are each sequences of nonnegative integers of length n. 2 on page 42. 7 on page 14). 16.

V G H o m ^ j ^ . n , C 2 ), then <£n sends w ® v t o ( c i 0 c 2 —• V r n ((w®v)(ci®c 2 ))) / where ci G Ci,c 2 G C 2 and 7 n : C^ 0 C £ —> (Ci ® Cf2)n is the map that shuffles the factors of together. 6. Let C\ and C 2 be m-coalgebras with structure maps f[Ci\n: Ci —> Hom^5n(9tl[C2]n j Cf), i = 1,2 and 0 < n < oo. ) ® Hom z s „(«[C 2 ] n , C2") ^ Homz5 n («[Ci]„ ® 9t[C 2 ]„, (Ci ® C 2 ) n ) JUSTIN R. SMITH 24 Remarks. 1. 5. 2. It is clear that the tensor product of two weakly coherent m-coalgebras is also weakly coherent.

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