Let
(
A
,
d
)
(A,d)
denote a free
r
r
-reduced differential graded
R
R
-algebra, where
R
R
is a commutative ring containing
n
−
1
{n^{ - 1}}
for
1
≤
n
>
p
1 \leq n > p
. Suppose a “diagonal”
ψ
:
A
→
A
⊗
A
\psi :A \to A \otimes A
exists which satisfies the Hopf algebra axioms, including cocommutativity and coassociativity, up to homotopy. We show that
(
A
,
d
)
(A,d)
must equal
U
(
L
,
δ
)
U(L,\delta )
for some free differential graded Lie algebra
(
L
,
δ
)
(L,\delta )
if
A
A
is generated as an
R
R
-algebra in dimensions below
r
p
rp
. As a consequence, the rational singular chain complex on a topological monoid is seen to be the enveloping algebra of a Lie algebra. We also deduce, for an
r
r
-connected CW complex
X
X
of dimension
≤
r
p
\leq rp
, that the Adams-Hilton model over
R
R
is an enveloping algebra and that
p
th
p\text {th}
powers vanish in
H
~
∗
(
Ω
X
;
Z
p
)
{\tilde H^ * }(\Omega X;{{\mathbf {Z}}_p})
.