150
6 Operads
The second convention would follow from the Koszul sign rule if we apply the
morphisms from the right. Equation (6.85) then reads as
(u ⊗ v)(f ⊗ g) = (−1)
|f ||v| f (u) ⊗ g(v)
and the unexpected sign comes from the commuting f over v. We denote, only for
the purposes of this remark, the first monoidal structure by ⊗ S and second by ⊗ M
(K abbreviating standard and M Montreal). We denote the corresponding monoidal
categories by Lin S and Lin M , respectively.
The two monoidal structures on Lin are related as follows. For any monoidal
category M with the monoidal product one can equip the same M by the opposite
monoidal structure † defined on objects by A † B := A B and similarly on
morphisms; let us denote M with the opposite monoidal structure by M † . It turns out
that the categories Lin
†
M and Lin S are isomorphic. The isomorphism is the identity of
the underlying categories, and the transformation turning ⊗
†
M into ⊗ S is the family
of maps
Φ U,V : U ⊗
†
M V = V ⊗ U → U ⊗ V = U ⊗ S V
given by Φ U,V (v ⊗ u) := v ⊗ u for u ∈ U and v ∈ V . 13
The category Lin with both monoidal structures is enriched over itself. For
homogeneous maps f : A → V and g : W → B, the corresponding enriched
functors
f
#
: Lin(V , W ) → Lin(A, W ) and g # : Lin(V , W ) → Lin(V , B)
are given, for ϕ ∈ Lin(V , W ), by
f
# (ϕ) := (−1)
|f ||ϕ| ϕ ◦ f and g # (ϕ) := ϕ ◦ g.
One can easily check that the obvious canonical isomorphism
Lin
A ⊗ B, C
∼ = Lin
A, Lin(B, C)
is functorial in B for both ⊗ = ⊗ S and ⊗ = ⊗ M .
Since odd modular operads possess operations of odd degrees, the form of
their axioms evaluated at concrete elements may depend on the chosen monoidal
structure of Lin. Let us, for instance, evaluate axiom (6.80) at homogeneous
elements
x ∈ T
S 1 {a}; g 1
, y ∈ T
S 2 {b, c}; g 2
and z ∈ T
S 3 {d}; g 3
13 Notice there are no signs!
6 Operads
The second convention would follow from the Koszul sign rule if we apply the
morphisms from the right. Equation (6.85) then reads as
(u ⊗ v)(f ⊗ g) = (−1)
|f ||v| f (u) ⊗ g(v)
and the unexpected sign comes from the commuting f over v. We denote, only for
the purposes of this remark, the first monoidal structure by ⊗ S and second by ⊗ M
(K abbreviating standard and M Montreal). We denote the corresponding monoidal
categories by Lin S and Lin M , respectively.
The two monoidal structures on Lin are related as follows. For any monoidal
category M with the monoidal product one can equip the same M by the opposite
monoidal structure † defined on objects by A † B := A B and similarly on
morphisms; let us denote M with the opposite monoidal structure by M † . It turns out
that the categories Lin
†
M and Lin S are isomorphic. The isomorphism is the identity of
the underlying categories, and the transformation turning ⊗
†
M into ⊗ S is the family
of maps
Φ U,V : U ⊗
†
M V = V ⊗ U → U ⊗ V = U ⊗ S V
given by Φ U,V (v ⊗ u) := v ⊗ u for u ∈ U and v ∈ V . 13
The category Lin with both monoidal structures is enriched over itself. For
homogeneous maps f : A → V and g : W → B, the corresponding enriched
functors
f
#
: Lin(V , W ) → Lin(A, W ) and g # : Lin(V , W ) → Lin(V , B)
are given, for ϕ ∈ Lin(V , W ), by
f
# (ϕ) := (−1)
|f ||ϕ| ϕ ◦ f and g # (ϕ) := ϕ ◦ g.
One can easily check that the obvious canonical isomorphism
Lin
A ⊗ B, C
∼ = Lin
A, Lin(B, C)
is functorial in B for both ⊗ = ⊗ S and ⊗ = ⊗ M .
Since odd modular operads possess operations of odd degrees, the form of
their axioms evaluated at concrete elements may depend on the chosen monoidal
structure of Lin. Let us, for instance, evaluate axiom (6.80) at homogeneous
elements
x ∈ T
S 1 {a}; g 1
, y ∈ T
S 2 {b, c}; g 2
and z ∈ T
S 3 {d}; g 3
13 Notice there are no signs!