2-dimensional topological quantum field theory and frobenius algebras

In this project, we define two dimensional topological quantum field theories (TQFTs) and show the relation between Frobenius algebras. More precisely, equivalence between the category 2dTQFT_{k}, of symmetric monoidal functors from the category, 2Cob, of two-dimensional cobordisms to the category, Vect_{k}, of vector spaces over a field and the category of, cFA_{k}, of commutative Frobenius algebras will be demonstrated. The work begins with basic properties of category theory. Moreover, a symmetric monoidal categories are explained. In the next three chapters, the category of cobordisms are constructed to define Frobenius algebras clearly. The category of Frobenius algebras are shown by the cobordism notation. At the end of the work, we use Atiyah axioms to explore TQFTs. His comment makes a relation between cobordisms and Frobenius algebras.