  This thesis provides an introduction to the various category theory ideas
employed in topological quantum field theory. These theories are viewed as
symmetric monoidal functors from topological cobordism categories into the
category of vector spaces. In two dimensions, they are classified by Frobenius
algebras. In three dimensions, and under certain conditions, they are
classified by modular categories. These are special kinds of categories in
which topological notions such as braidings and twists play a prominent role.
There is a powerful graphical calculus available for working in such
categories, which may be regarded as a generalization of the Feynman diagrams
method familiar in physics. This method is introduced and the necessary
algebraic structure is graphically motivated step by step.
  A large subclass of two-dimensional topological field theories can be
obtained from a lattice gauge theory construction using triangulations. In
these theories, the gauge group is finite. This construction is reviewed, from
both the original algebraic perspective as well as using the graphical calculus
developed in the earlier chapters.
  This finite gauge group toy model can be defined in all dimensions, and has a
claim to being the simplest non-trivial quantum field theory. We take the
opportunity to show explicitly the calculation of the modular category arising
from this model in three dimensions, and compare this algebraic data with the
corresponding data in two dimensions, computed both geometrically and from
triangulations. We use this as an example to introduce the idea of a quantum
field theory as producing a tower of algebraic structures, each dimension
related to the previous by the process of categorification.
