Chapter 4 - Operads

Summary

We introduce the notion of dispecies relative to a fixed hyperplane arrangement. The category of dispecies carries a monoidal structure which we call the substitution product. Operads are monoids in this monoidal category. We describe the free operad on a dispecies, and then proceed to operad presentations with an emphasis on binary quadratic operads. Apart from the substitution product, the category of dispecies also carries the Hadamard product which turns it into a 2-monoidal category. Hopf operads are bimonoids in this 2-monoidal category. We use these ideas to construct the black and white circle products on binary quadratic operads. We discuss three main examples of operads, namely, commutative, associative, Lie. These are all binary quadratic. Further, under a suitable notion of quadratic duality, the commutative and Lie operads are duals of each other, while the associative operad is self-dual. These can be viewed as extensions of well-known facts from the classical theory of May operads. The category of species is a left module category over the monoidal category of dispecies (under the substitution product). Hence, to each operad, one can associate the category of its left modules. A left module over the associative operad is the same as a monoid in species, over the commutative operad is the same as a commutative monoid in species, over the Lie operad is the same as a Lie monoid in species. To every operad, one can attach an (associative) algebra called its incidence algebra. The incidence algebra of the commutative operad is the flat-incidence algebra, of the associative operad is the lune-incidence algebra, and of the Lie operad is the Tits algebra. The incidence algebra of any connected quadratic operad is elementary and its quiver can be explicitly described. Operads can also be defined in the more general setting of left regular bands. Interestingly, the commutative, associative, Lie operads extend to this setting.