On the Operads of J . P . May

Author's Note. When this manuscript was submitted in January 1972, the editor asked that it be expanded to study the relation of operads to clubs. The author found this too daunting a task at a busy time and the manuscript was never published. Reading through the manuscript now, more than thirty years later, elicits two strong impressions. First, the treatment is very complete: the only item not discussed in detail is the coherence of the monoidal structure given by the functor T • S on [P, V]. Secondly, it was done—for instance in proving the associativity (R • T) • S ∼ = R • (T • S)—with bare hands. Today one could argue as follows, using universal properties; the author learned this approach from Aurelio Carboni. P op , which is in fact isomorphic to P, is the free symmetric monoidal category on 1. So to give an object of [P, V], or a functor T : 1 → [P, V], is equally to give a strong monoidal functor P op → [P, V], where the latter has the convolution monoidal structure ⊗; this is the strong monoidal functor sending m to the tensor power T m = T ⊗T ⊗.. .⊗T. By Theorem 5.1 of [12], this is equally to give a cocontinuous strong monoidal functor T : [P, V] → [P, V]; this is the left Kan extension − • T , and T is recovered from T as T (J) = J • T. Now the desired associativity (− • T) • S ∼ = − • (T • S) is just the associativity of these cocontinuous strong monoidal functors. I am grateful to my colleagues Lack, Street, and Wood for suggesting this article for the TAC Reprint series, and to Flora Armaghanian for producing the LaTeX version.