Let EK be the simplicial suspension of a pointed simplicial set K. We construct a chain model of the James map, αK: CK → ΩCEK. We compute the cobar diagonal on ΩCEK, not assuming that EK is 1-reduced, and show that αK is comultiplicative. As a result, the natural isomorphism of chain algebras TCK → ΩCK preserves diagonals.
In an appendix, we show that the Milgram map, Ω(A ⊗ B) → ΩA ⊗ ΩB, where A and B are coaugmented coalgebras, forms part of a strong deformation retract of chain complexes. Therefore, it is a chain equivalence even when A and B are not 1-connected.Homology, Homotopy and Applications, Vol. 9 (2007), No. 2, pp.209-231.