Definitions: Operads, Algebras and Modules

Let S be a symmetric monoidal category with product ⊗ and unit object κ. Definition 1. An operad C in S consists of objects C (j), j ≥ 0, a unit map η : κ → C (1), a right action by the symmetric group Σj on C (j) for each j, and product maps γ : C (k)⊗ C (j1)⊗ · · · ⊗ C (jk) → C (j) for k ≥ 1 and js ≥ 0, where ∑ js = j. The γ are required to be associative, unital, and equivariant in the following senses. (a) The following associativity diagrams commute, where ∑ js = j and ∑ it = i; we set gs = j1 + · · ·+ js, and hs = igs−1+1 + · · ·+ igs for 1 ≤ s ≤ k: