6.5 Odd Modular Operads
149
defined for arbitrary finite disjoint sets S 1 , S 2 , symbols a, b, genera g 1 , g 2 ∈ A, and
degree +1 contractions
• uv = • vu : T
S {u, v}; g
→ T (S; g + s)
(6.79)
given for any finite set S, genus g ∈ A, and symbols u, v. These data are required
to satisfy axioms of Definition 6.16 for the operations a ◦ b and ◦ uv , with the only
difference that the formulas in axioms (iv)–(vii) acquire the minus signs, i.e., read
as
a • b (1 ⊗ c • d ) = − c • d ( a • b ⊗1),
(6.80)
• ab • cd = − • cd • ab ,
(6.81)
• ab c • d = − • cd a • b , and
(6.82)
a • b (• uv ⊗ 1) = − • uv a • b .
(6.83)
A morphism of odd modular operads is a morphism of the underlying modular
modules commuting with all structure operations.
The minus signs in (6.80)–(6.83) are forced by the Koszul sign conventions,
as both the compositions and contractions are “objects” of degree +1. Informally,
odd modular operads are modular operads whose structure operations have “wrong”
degrees and also the signs of some of the axioms are “wrong.” Because of nontrivial
signs and degrees, odd modular operads, unlike the ordinary ones, do not exist in an
arbitrary symmetric monoidal category but, e.g., in symmetric monoidal categories
enriched over graded vector spaces.
Remark 6.11 The category of Lin of Z-graded vector spaces and their homogeneous
linear maps of arbitrary degrees has two symmetric monoidal structures, the standard one and the one which we call, from reasons which will became clear later, the
Montreal monoidal structure. The monoidal product of objects is for both structures
the standard tensor product of graded vector spaces, but the structures differ by their
actions on morphisms. The prevailing convention is that, for homogeneous maps
f : V → W , g : V → W and homogeneous vectors u ∈ V , v ∈ W one
defines
(f ⊗ g)(u ⊗ v) = (−1)
|g||u| f (u) ⊗ g(v),
(6.84)
while some categorists at McGill University in Montreal would prefer
(f ⊗ g)(u ⊗ v) = (−1)
|f ||v| f (u) ⊗ g(v).
(6.85)
149
defined for arbitrary finite disjoint sets S 1 , S 2 , symbols a, b, genera g 1 , g 2 ∈ A, and
degree +1 contractions
• uv = • vu : T
S {u, v}; g
→ T (S; g + s)
(6.79)
given for any finite set S, genus g ∈ A, and symbols u, v. These data are required
to satisfy axioms of Definition 6.16 for the operations a ◦ b and ◦ uv , with the only
difference that the formulas in axioms (iv)–(vii) acquire the minus signs, i.e., read
as
a • b (1 ⊗ c • d ) = − c • d ( a • b ⊗1),
(6.80)
• ab • cd = − • cd • ab ,
(6.81)
• ab c • d = − • cd a • b , and
(6.82)
a • b (• uv ⊗ 1) = − • uv a • b .
(6.83)
A morphism of odd modular operads is a morphism of the underlying modular
modules commuting with all structure operations.
The minus signs in (6.80)–(6.83) are forced by the Koszul sign conventions,
as both the compositions and contractions are “objects” of degree +1. Informally,
odd modular operads are modular operads whose structure operations have “wrong”
degrees and also the signs of some of the axioms are “wrong.” Because of nontrivial
signs and degrees, odd modular operads, unlike the ordinary ones, do not exist in an
arbitrary symmetric monoidal category but, e.g., in symmetric monoidal categories
enriched over graded vector spaces.
Remark 6.11 The category of Lin of Z-graded vector spaces and their homogeneous
linear maps of arbitrary degrees has two symmetric monoidal structures, the standard one and the one which we call, from reasons which will became clear later, the
Montreal monoidal structure. The monoidal product of objects is for both structures
the standard tensor product of graded vector spaces, but the structures differ by their
actions on morphisms. The prevailing convention is that, for homogeneous maps
f : V → W , g : V → W and homogeneous vectors u ∈ V , v ∈ W one
defines
(f ⊗ g)(u ⊗ v) = (−1)
|g||u| f (u) ⊗ g(v),
(6.84)
while some categorists at McGill University in Montreal would prefer
(f ⊗ g)(u ⊗ v) = (−1)
|f ||v| f (u) ⊗ g(v).
(6.85)