6.5 Odd Modular Operads
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i.e., calculate
a • b (1 ⊗ c • d )(x ⊗ y ⊗ z) = − c • d ( a • b ⊗1)(x ⊗ y ⊗ z).
While in Lin S we get
(−1)
|x| x a • b (y c • d z) = −(x a • b y) c • d z,
(6.86)
in Lin M we obtain
x a • b (y c • d z) = −(−1)
|z| (x a • b y) c • d z.
(6.87)
Likewise, axiom (6.83) in Lin S reads
• uv (x) a • b y = − • uv (x a • b y),
(6.88)
while in Lin M one would get
(−1)
|y|
• uv (x) a • b y = − • uv (x a • b y)
(6.89)
for x, y belonging to the appropriate components of T . Also the derivation property
of the differential with respect to the a • b -operation depends on the chosen monoidal
structures. In Lin S it reads
d(x a • b y) = −(dx) a • b y − (−1)
|x| x a • b (dy),
while in Lin M it is
d(x a • b y) = −(−1)
|y| (dx) a • b y − x a • b (dy).
Axioms (6.81) and (6.82) are the same in both monoidal structures.
It fortunately turns out that the categories of odd modular operads in Lin S and in
Lin S are isomorphic. Indeed, we leave as an exercise to prove that the modification
x a • b y → (−1)
|x|+|y| x a • b y, • uv (x) → (−1)
|x|
• uv (x), d(x) → (−1)
|x| d(x),
(6.90)
turns an odd modular operad in Lin S into one in Lin M and vice versa. If not stated
otherwise, all odd modular operads will be considered in Lin with the standard
monoidal structure. For that reason we drop the subscript S.