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6 Operads
y
z
x
a
p
q
r
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u
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0
2 1
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Fig. 6.1 A tree-like diagram of correlation functions and propagators
We also need a rule that would identify two blobs that differ only by relabeling the
inputs. This is abstractly expressed by requiring an action on, e.g., P
{p, q, r, a}
by the group of permutations of the set {p, q, r, a}.
Cyclic operads are abstractions of structures of blobs and propagators for the
cases when the related diagrams are simply connected, i.e., when they do not have
loops. The general case is covered by the notion of modular operads discussed in
Sect. 6.4.
Let us proceed to a precise definition. Denote by Cor the category of finite sets
and their isomorphisms. Finite sets in Cor will serve as indexing sets for the inputs
of abstract blobs. In this context we call Cor the category of corollas, whence the
notation. As usual, we denote by A 1 A 2 the union of disjoint (finite) sets A 1 and
A 2 . Two morphisms ρ : A 1 → B 1 , σ : A 2 → B 2 give the obvious induced map
ρ σ : A 1 A 2 → B 1 B 2 .
With this terminology, we may formulate
Definition 6.1 A cyclic operad P is a family
P =
P(S) ∈ Chain | S ∈ Cor
of dg-vector spaces together with degree 0 morphisms
P(ρ) : P(S) → P(T )
(6.1)