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3) R E D U C E D BAR C O N S T R U C T I O N S ON DERHAM COMPLEXES 27 Corollary 1. Under the same hypothesis as in the theorem, if X is a simply connected topological space, then H*(B(A)) % Η*(ΩΧ; k). For the nonsimply connected case, we mention the next result, which depends on the work [5]. ) Corollary 2. U n d e r the same hypothesis as in the theorem, if the fundamental group G = π^Χ, x0) is finitely generated and if ε: kG -• k is the augmentation of the group algebra kG, then H°(B(A)) is naturally isomorphic with the Hopf algebra of all /c-valued linear functional on kG that annul some power of the augmentation ideal ker ε.

The components of aff C are filtered if a n d only if every diagram in C of the form A can be completed to a (not necessarily commutative) square in C, and for each diagram of aff C of the form A there is an element β of © such that β<χ = β'α'. c e | C | ZC(£, C) with coefficients summing to one Proof. If β = £ rtßi a n d ßoc = /fa', then using the first assumption we can choose, for each object in C that appears as the codomain of a ßi9 a morphism y with c o m m o n codomain. Composing each /}, with the a p p r o priate y does not alter the validity of ßoc = βα', but now β will be a morphism of aff C.

79 (1973), 1033-1035. Κ. T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 8 3 - 9 8 . Κ. T. Chen, Connections, h o l o n o m y and p a t h space homology, Proc. , Amer. Math. Soc. 27 (1975), 39-52. S. Eilenberg a n d S. M a c Lane, O n the g r o u p s of H(U, n), I, Ann. of Math. 58 (1953), 55-106. S. Eilenberg a n d S. M a c Lane, O n the g r o u p s of Η(Π, η), II, Ann. of Math. 60 (1954), 49-139. S. Eilenberg a n d J. C. M o o r e , H o m o l o g y a n d fibrations I, Comment.

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