
72 Algebraic Operads: An Algorithmic Companion
A nonsymmetric operad is a nonsymmetric collection of vector spaces
P = {P(n)}
n≥0
equipped with an element id ∈ P(1) and maps
◦
i
: P(n) ⊗P(m) → P(n + m − 1), α ⊗ β 7→ α ◦
i
β
which satisfy the following properties for all α ∈ P(n), β ∈ P(m), γ ∈ P(r):
• sequential axiom:
(α ◦
i
β) ◦
j
γ = α ◦
i
(β ◦
j−i+1
γ) for i ≤ j ≤ i + m −1; (3.3)
• parallel axiom:
(α ◦
i
β) ◦
j
γ =
(
(α ◦
j−m+1
γ) ◦
i
β, i + m ≤ j ≤ n + m − 1,
(α ◦
j
γ) ◦
i+r−1
β, 1 ≤ j ≤ i − 1;
(3.4)
• unit axiom:
id ◦
1
α = α, α ◦
i
id = α for 1 ≤ i ≤ n. (3.5)
The following result is well known; it is featured in every textbook on
operads. Throughout this book, we will furnish some ...