The goal of this monograph is to develop Hopf theory in the setting of a
real reflection arrangement. The central notion is that of a Coxeter
bialgebra which generalizes the classical notion of a connected graded
Hopf algebra. The authors also introduce the more structured notion of a
Coxeter bimonoid and connect the two notions via a family of functors
called Fock functors. These generalize similar functors connecting Hopf
monoids in the category of Joyal species and connected graded Hopf
algebras. This monograph opens a new chapter in Coxeter theory as well
as in Hopf theory, connecting the two. It also relates fruitfully to
many other areas of mathematics such as discrete geometry, semigroup
theory, associative algebras, algebraic Lie theory, operads, and
category theory. It is carefully written, with effective use of tables,
diagrams, pictures, and summaries. It will be of interest to students
and researchers alike.