  In recent years, there has been great interest in the study of
categorification, specifically as it applies to the theory of quantum groups.
In this thesis, we would like to provide a new approach to this problem by
looking at Hall algebras. It is know, due to Ringel, that a Hall algebra is
isomorphic to a certain quantum group. It is our goal to describe a
categorification of Hall algebras as a way of doing so for their related
quantum groups. To do this, we will take the following steps. First, we
describe a new perspective on the structure theory of Hall algebras. This view
solves, in a unique way, the classic problem of the multiplication and
comultiplication not being compatible. Our solution is to switch to a different
underlying category $\Vect^K$ of vector spaces graded by a group $K$ called the
Grothendieck group. We equip this category with a nontrivial braiding which
depends on the $K$-grading. With this braiding and a given antipode, we find
that the Hall algebra does become a Hopf algebra object in $\Vect^K$. Second,
we will describe a categorification process, call `groupoidification', which
replaces vector spaces with groupoids and linear operators with `spans' of
groupoids. We will use this process to construct a braided monoidal bicategory
which categorifies $\Vect^K$ via the groupoidification program. Specifically,
graded vector spaces will be replaced with groupoids `over' a fixed groupoid
related to the Grothendieck group $K$. The braiding structure will come from an
interesting groupoid $\EXT(M,N)$ which will behave like the Euler
characteristic for the Grothendieck group $K$. We will finish with a
description of our plan to, in future work, apply the same concept to the
structure maps of the Hall algebra, which will eventually give us a Hopf
2-algebra object in our braided monoidal bicategory.
