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6 Operads
Definition 6.2 A morphism Φ : P → Q of cyclic operads is a collection
Φ = {Φ S : P(S) → Q(S) | S ∈ Cor}
of degree 0 morphisms of dg-vector spaces that commute with all structure
operations. This means that for ρ : S → T as in (6.1) the diagram
Φ T
(ρ)
(ρ)
Φ S
(T )
(T )
(S)
(S)
commutes as does, for S 1 , S 2 , a, b as in (6.2), the diagram
a ◦ b
Φ S 1 S 2
1 a} ⊗ Φ S 2 b}
a ◦ b
(S 1 S 2 ) .
S 1 a} ⊗
S 2 b}
(S 1 S 2 )
S 1 a} ⊗
S 2 b}
We denote by CycOp the category of cyclic operads and their morphisms.
Remark 6.1 It should be clear that P(S) is an abstraction of the space of blobs with
inputs indexed by the elements of the finite set S. Axiom (i) of Definition 6.1 says
that the rule S → P(S), ρ → P(ρ) defines a covariant functor from the category
Cor to the category Chain of dg-vector spaces and their degree 0 morphisms.
Axiom (ii) describes the behavior of contractions with respect to reindexations.
Axiom (iii) says that, for any x ∈ P
S 1 {a}
and y ∈ P
S 2 {b}
,
a ◦ b (x ⊗ y) = (−1)
|x||y|
b ◦ a (y ⊗ x).
If we write x a ◦ b y instead of a ◦ b (x ⊗ y) and similarly for b ◦ a (y ⊗ x), we see
that (iii) is the graded commutativity
x a ◦ b y = (−1)
|x||y| y b ◦ a x
of the contractions. Informally this means that the results of the contractions of
the edges in Fig. 6.1 do not depend, modulo the Koszul sign, on their orientations.
Likewise, axiom (iv) requiring that
a ◦ b (1 ⊗ c ◦ d )(x ⊗ y ⊗ z) = c ◦ d ( a ◦ b ⊗1)(x ⊗ y ⊗ z)