Categorification is a term that was coined by Louis Crane and Igor
Frenkel to describe the process of refounding mathematical theories
from their bases in set theory to category theoretical analogs. In
this thesis we provide a brief review of the theory of complex linear
representations of the dihedral groups, provide the needed
category theory background to define and discuss abelian categories
and related structures, present the basic notions of algebraic
categorification especially as it relates to representation theory,
and finally detail how we can construct, and fail to construct,
certain categorifications of the simple representations of the
dihedral group D3.