We first introduce some concepts in category theory, defining monoidal categories, functors, natural transformations, monoids, and comonoids. We also equip monoidal categories with symmetric structure. Then, we prove a theorem describing the universality of the monoidal category of finite ordinals. Next, we define cobordisms between oriented manifolds and construct the symmetric monoidal category of n-cobordisms. We determine the generators of the category of 2-cobordisms and find a sufficient set of relations. Continuing with our treatment of monoidal categories, we define pairings and Frobenius objects. We then characterize the Frobenius property with pairings. Using this, we prove the theorem that describes the universality of the symmetric monoidal category of 2-cobordisms. We consider a special case of this theorem for the category of vector spaces. We define topological quantum field theories and Frobenius algebras, and then provide a more specific description of non-degenerate pairings. Finally, we explore some examples of topological quantum field theories.
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