6.1 Cyclic Operads
107
modulo the identifications
v ω(1) ⊗ · · · ⊗ v ω(n) ∼ (σ ) v ωσ (1) ⊗ · · · ⊗ v ωσ (n) , σ ∈ Σ n ,
where (σ ) is the Koszul sign ((1) in Part II) of the permutation σ .
Remark 6.7 The need for a subtler version of the tensor product is caused by the fact
that the category dgVec of dg-vector spaces is a non-strict symmetric monoidal
category. Similar unordered products can be defined in any symmetric monoidal
category with finite colimits, see, e.g., [12, Def. II.1.58].
Let us formulate two important properties of unordered tensor products.
Lemma 6.1 Let σ : S → D be an isomorphism of finite sets, {V c } c∈S and {W d } d∈D
collections of graded vector spaces, and ϕ = {ϕ c : V c → W σ c } c∈S a family of linear
maps. Then the assignment
c∈S
V c
v ω(1) ⊗ · · · ⊗ v ω(n)
−→
w σ ω(1) ⊗ · · · ⊗ w σ ω(n)
∈
d∈D
W d
with w σ ω(i) := ϕ ω(i) (v ω(i) ) ∈ W σ ω(i) , 1 ≤ i ≤ n, defines a natural map
(σ, ϕ) :
c∈S
V c →
d∈D
W d
of unordered products
Proof. A direct verification.
A particularly important case of the above lemma is when V c = V d = V for all
c ∈ S, d ∈ D, and ϕ c : V → V is the identity for all c ∈ S. Lemma 6.1 then gives
a natural map
σ := (σ, ϕ) :
c∈S
V c →
d∈D
V d .
(6.14)
Lemma 6.2 For disjoint finite sets S , S , one has a canonical isomorphism
c ∈S
V c ⊗
c ∈S
V c ∼ =
c∈S S
V c .