  A vector species is a functor from the category of finite sets with
bijections to vector spaces; informally, one can view this as a sequence of
$S_n$-modules. A Hopf monoid (in the category of vector species) consists of a
vector species with unit, counit, product, and coproduct morphisms satisfying
several compatibility conditions, analogous to a graded Hopf algebra. We say
that a Hopf monoid is strongly linearized if it has a "basis" preserved by its
product and coproduct in a certain sense. We prove several equivalent
characterizations of this property, and show that any strongly linearized Hopf
monoid which is commutative and cocommutative possesses four bases which one
can view as analogues of the classical bases of the algebra of symmetric
functions. There are natural functors which turn Hopf monoids into graded Hopf
algebras, and applying these functors to strongly linearized Hopf monoids
produces several notable families of Hopf algebras. For example, in this way we
give a simple unified construction of the Hopf algebras of superclass functions
attached to the maximal unipotent subgroups of three families of classical
Chevalley groups.
