6.1 Cyclic Operads
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(iv) For disjoint finite sets S 1 , S 2 , S 3 and a ∈ S 1 , b, c ∈ S 2 , b = c, d ∈ S 3 , one has
the equality
a ◦ b (1 ⊗ c ◦ d ) = c ◦ d ( a ◦ b ⊗1)
of maps P(S 1 ) ⊗ P(S 2 ) ⊗ P(S 3 ) → P
S 1 S 2 S 3 \ {a, b, c, d}
.
The category Cor of finite sets is equivalent to its full skeletal subcategory whose
objects are the sets [n] := {1, . . . , n}, n ≥ 0, with [0] interpreted as the empty set ∅.
It is therefore not surprising that there exists a skeletal version of Definition 6.1 in
which the components of cyclic operads are not indexed by arbitrary finite sets, but
by the finite ordinals [n], n ≥ 1. It can be obtained as follows.
For a non-negative integer n denote P(n) := P
[n]
. The definition of the
skeletal versions of the a ◦ b -operations (6.2) involves, for m, n ≥ 0, 1 ≤ i ≤ m + 1,
1 ≤ j ≤ n + 1, an isomorphism
κ = κ ij :
[m + 1] \ {i}
[n + 1] \ {j }
∼ =
−→ [m + n]
(6.7)
given as follows. For a ∈ [m + 1] \ {i} put
κ(a) :=
a,
1 ≤ a < i,
a + n − 1, i < a ≤ m + 1,
while for b ∈ [n + 1] \ {j },
κ(b) :=
b − j + i + n, 1 ≤ b < j,
b − j + i − 1, j < b ≤ n + 1.
With these conventions, define
i ◦ j : P(m + 1) ⊗ P(n + 1) → P(m + n), 1 ≤ i ≤ m + 1, 1 ≤ j ≤ n + 1,
as the composition
P(m + 1) ⊗ P(n + 1) =P([m + 1]) ⊗ P([n + 1])
i ◦ j
−−−→
(6.8)
P
([m + 1] \ {i}) ([n + 1] \ {j })
P(κ)
−−−→ P([m + n]) = P(m + n),
where i ◦ j is the extended operation (6.6) for S 1 = [m+1]\{i} and S 2 = [n+1]\{j }.
Notice finally that each P(n) bears a natural right action of the symmetric group
Σ n = Aut [n] .