6.4 Modular Operads
141
Observe that
card(QO(S, G)) < ∞
for any (S, G) ∈ S. The reader can easily verify that:
Theorem 6.1 QO is a stable modular operad in the category of sets.
As shown in [10], QO is the modular completion of the non-Σ cyclic operad Ass
from Example 6.17 in a certain category of non-Σ modular operads. The relation
between QO and Ass is therefore similar as the relation between QC and Com
described in Example 6.26.
Example 6.28 Let V be a graded vector space and s ∈ V ⊗V a symmetric degree 0
tensor. We are going to define a modular extension of the cyclic endomorphism
operad from Example 6.6. For a finite set S and g ∈ A we put
End V (S; g) := Lin
c∈S V c , k
=
c∈S V c
#
(6.61)
so End V (S; g) equals the component End V (S) of the cyclic endomorphism operad
for each g ∈ A. The action of isomorphisms of finite sets and also the a ◦ b -operations
are defined precisely as in Example 6.6. The contraction ◦ uv f ∈ End V (S; g + s) of
f ∈ End V
S {u, v}; g
is the composition
c∈S
V c ∼ = k ⊗
c∈S
V c
s⊗1
−−−→ V u ⊗ V v ⊗
c∈S
V c
∼ =
−→
c∈S{u,v}
V c
f
−→ k, (6.62)
where s is interpreted as a degree 0 map k → V u ⊗ V v . It follows from the
commutativity of the diagram
⊗ s
∼ =
∼ =
s ⊗
∼ =
∼ =
c∈S V c ⊗
c∈S V c ⊗ V u ⊗ V v
c∈S u,v} V c
V u ⊗ V v ⊗ c∈S V c
⊗ c∈S V c
c∈S V c
(6.63)