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6 Operads
Let us describe the compositions (6.2). Thanks to equality (6.45), every element of
Ass
S 1 {a}
can be (uniquely) represented as (c
1 , . . . , c
n , a) and, likewise, each
element of Ass
S 2 {b}
can be represented as (b, c
1 , . . . , c
m ). We then have
(c
1 , . . . , c
n , a) a ◦ b (b, c
1 , . . . , c
m ) := (c
1 , . . . , c
n , c
1 , . . . , c
m ).
Example 6.19 The category of stable non-Σ-cyclic operads in the category Set of
sets has the terminal object ∗ cyclic given by
∗ cyclic (S) :=
∗ if card(S) ≥ 3 and
∅ otherwise,
where ∗ is a one-point set, with all structure operations the isomorphisms ∗×∗
∼ =
− → ∗.
The non-Σ cyclic operad Ass of Example 6.17 is the linearization of this operad,
Ass ∼ = Span(∗ cyclic ).
Example 6.20 As the terminal stable cyclic operad ∗ cyclic of Example 6.2 has an
interpretation in terms of oriented genus zero surfaces with holes, the operad ∗ cyclic
is isomorphic with the operad W 0 whose components W 0 (S) for card(S) ≥ 3,
consist of isomorphism classes of planar cogwheels
whose teeth are indexed by the cyclically ordered set S, with the cyclic order
agreeing with the one induced by the anti-clockwise orientation of the plane. If
card(S) ≤ 2 we put W 0 (S) := ∅.
The structure operations are induced by gluing these cogwheels together along
the tips of their teeth so that the orientation is preserved: