112
6 Operads
with v, w ∈
c∈S V c . The symmetry of s resp. of B implies that, if we take (1 ⊗ f )
instead of (f ⊗ 1) in (6.19) resp. w ⊗ v instead of v ⊗ w in (6.20), the resulting
maps will be the same. The following lemma is easy to prove.
Lemma 6.3 The family Φ = {Φ S } is a morphism of operads if and only if
s = (1 V ⊗ B ⊗ 1 V )(s ⊗ s).
(6.21)
Likewise, Ψ = {Ψ S } is a morphism of operads if and only if
B = (B ⊗ B)(1 V ⊗ s ⊗ 1 V ).
(6.22)
Remark 6.8 Equations (6.21) resp. (6.22) have simple geometric expressions. If we
depict B : V ⊗ V → k as an abstract operation with two inputs and no output, 2 i.e.,
B =
and s as an operation with no input and two outputs, i.e.,
s =
,
then (6.21) is expressed as
=
while (6.22) as
=
Recall that B is non-degenerate if, for each x ∈ V , there exists y ∈ V such that
B(x, y) = 0. It is a standard fact that this condition is equivalent to the existence of
a (necessarily unique) symmetric s ∈ V ⊗ V such that
(1 V ⊗ B)(s ⊗ u) = (B ⊗ 1 V )(u ⊗ s) = u,
(6.23)
for each u ∈ V or, in pictures in the spirit of Remark 6.8,
=
=
2 Since k is the unit of the monoidal category dgVec, it does not count as an output.