6.2 Non-Σ Cyclic Operads
129
The action Ass(σ ) of an order-preserving isomorphisms of cyclically ordered set is
the identity, and the compositions
a ◦ b : Ass
S 1 {a}
⊗ Ass
S 2 {b}
−→ Ass
S 1 S 2
are the canonical isomorphisms. Then Ass with the above structure is a stable nonΣ cyclic operad.
The definition of the operad Ass looks very much the same as the definition of
Com given in Example 6.1, but one must remember that these operads belong to
different categories. The suboperads, ideals, presentations, and the related notions
for non-Σ cyclic operads can be defined analogously as for ordinary operads; we
leave the details to the reader. The following statement describes a presentation of
the non-Σ cyclic operad Ass.
Proposition 6.8 The non-Σ cyclic operad Ass is generated by an element
μ {1,2,3} ∈ Ass({1, 2, 3}) such that
μ {1,2,3} 3 ◦ 3 μ {3,4,5}
(6.44)
is cyclically symmetric in 1, 2, 4, 5.
Example 6.18 The symmetrization (6.42) Ass := Sym(Ass) has a nice explicit
description. For a set with n ≥ 3 elements one has
Ass(S) = Span
ω : {1, . . . , n}
∼ =
−→ S
∼
,
the vector space spanned by the set of isomorphisms ω between {1, . . . , n} and S,
modulo the relation ∼ that identifies, for each cyclic permutation λ ∈ Σ n , the
isomorphism ω with ωλ.
It is convenient to denote the equivalence class of [ω] by (c 1 , . . . , c n ), with c i :=
ω(i), 1 ≤ i ≤ n. By the definition of the equivalence, one has the equality
(c
1 , . . . , c
n ) = (c
1 , . . . , c
n )
(6.45)
if and only if c
i = c
λ(i) for some cyclic permutation λ ∈ Σ n , 1 ≤ i ≤ n.
In this language, the action (6.1) is given by
Ass(σ )(c 1 , . . . , c n ) = (σ c 1 , . . . , σ c n ).
129
The action Ass(σ ) of an order-preserving isomorphisms of cyclically ordered set is
the identity, and the compositions
a ◦ b : Ass
S 1 {a}
⊗ Ass
S 2 {b}
−→ Ass
S 1 S 2
are the canonical isomorphisms. Then Ass with the above structure is a stable nonΣ cyclic operad.
The definition of the operad Ass looks very much the same as the definition of
Com given in Example 6.1, but one must remember that these operads belong to
different categories. The suboperads, ideals, presentations, and the related notions
for non-Σ cyclic operads can be defined analogously as for ordinary operads; we
leave the details to the reader. The following statement describes a presentation of
the non-Σ cyclic operad Ass.
Proposition 6.8 The non-Σ cyclic operad Ass is generated by an element
μ {1,2,3} ∈ Ass({1, 2, 3}) such that
μ {1,2,3} 3 ◦ 3 μ {3,4,5}
(6.44)
is cyclically symmetric in 1, 2, 4, 5.
Example 6.18 The symmetrization (6.42) Ass := Sym(Ass) has a nice explicit
description. For a set with n ≥ 3 elements one has
Ass(S) = Span
ω : {1, . . . , n}
∼ =
−→ S
∼
,
the vector space spanned by the set of isomorphisms ω between {1, . . . , n} and S,
modulo the relation ∼ that identifies, for each cyclic permutation λ ∈ Σ n , the
isomorphism ω with ωλ.
It is convenient to denote the equivalence class of [ω] by (c 1 , . . . , c n ), with c i :=
ω(i), 1 ≤ i ≤ n. By the definition of the equivalence, one has the equality
(c
1 , . . . , c
n ) = (c
1 , . . . , c
n )
(6.45)
if and only if c
i = c
λ(i) for some cyclic permutation λ ∈ Σ n , 1 ≤ i ≤ n.
In this language, the action (6.1) is given by
Ass(σ )(c 1 , . . . , c n ) = (σ c 1 , . . . , σ c n ).