Chapter 2 - Species and bimonoids

Summary

We introduce the notion of species relative to a fixed hyperplane arrangement. Roughly speaking, a species is a family of vector spaces, one for each face of the arrangement, along with linear isomorphisms between vector spaces indexed by faces of the same support. Next, we introduce the notion of a monoid in species. It consists of a species equipped with "product'' maps from a vector space indexed by a face to a vector space indexed by a smaller face. These are subject to naturality, associativity, unitality axioms. There is also a dual notion of a comonoid in species defined using `"coproduct'' maps, and a mixed self-dual notion of a bimonoid in species. We also define commutativity for a monoid and dually cocommutativity for a comonoid. A bimonoid could be commutative, cocommutative, both or neither. Commutative monoids, cocommutative comonoids, bicommutative bimonoids are convenient to formulate using flats rather than faces. In addition to the above, we discuss related objects such as q-bimonoids (which include bimonoids, signed bimonoids, 0-bimonoids), signed commutative monoids, and partially commutative monoids. The latter interpolate between monoids and commutative monoids. The above notion of species when specialized to the braid arrangements relates to the classical notion of Joyal species.