A Quick Introduction to Operads

Operads (or at least An and En operads, which is what we’ll focus on) are a formalism for discussing various degrees to which associativity and commutativity can fail. Suppose I have a space X with a multiplication μ : X ×X → X that is not associative. In particular, μ(−, μ(−,−)) and μ(μ(−,−),−) are two different ternary operations. One approach is to consider a space O(3) of ternary operations, in which μ(μ(−,−),−) and μ(−, μ(−,−)) are two different points. Then the geometry of this space tells us something about exactly how badly associativity fails—strict associativity corresponds to this space being one point (so in particular μ(−, μ(−,−)) = μ(μ(−,−),−)), and the idea is that the next best thing is for the space to be contractible. For every n, we can also consider a space O(n) of n-ary operations, where the operations μ(μ(μ(. . . ),−),−) etc. are points. The best kind of associativity short of strict associativity is when all the O(n)’s are contractible; in this case, we say the multiplication is A∞.