There have been several exciting developments in the fields of higher categories, quantum, and topology. This paper tries to best give the classification of extended 2-dimensional topological theories, with a potential for various future developments and applications. A topological quantum field theory (TQFT) is a factor between two specific categories, an algebraic category and a geometric category. The geometric category is classified as a bordisms (Joel, 1994). There is another such category for every non negative integer, d, leading to resulting in a notion of d-dimensional TQFT. The algebraic category is normally a vector-space category over a fixed ground of field. The categories have other structures.
Systematic Monodidal Bicategories
The mathematical axiomatization of the systematic monoidal bicategories relates to the work of Voevodsky and Kapronov on braided monoidal 2-categories. This was further clarified in Day and Street work. It gives a clear explanation of how the categorification braided monoidal, and symmetric monoidal categories acquire an additional layer (Andezej, 1980). In addition there are two other results of about symmetric monoidal bicategories. First, there is a theorem simply characterizes those systematic monoidal functors, which also known as symmetric monoidal homomorphism.
Topological Field Theories and Planar Decompositions
The modern theoretical physics encounters an increasingly influential impact on mathematics over the last three decades. Particularly, the emphasis of other mathematician’s work on topological aspects of quantum theory of field has resulted in many significant developments in the two fields, where most of them continue to be more influential to this day.
In addition to the provision of a connection between geometry and algebra, TQFTs have at least two other significant functions. Firstly, it serves as toy models for several complex non-topological field theories. The study of the simple topological theories, there might be a slight hope of gleaming some insight or structure applicable to other general quantum field theories. As symmetric monoidal functor must send the monoidal unit of the category source, it is very important to understand the two third principles of topological quantum field theories.
However, the manifold invariants seem to appear to be more interesting in low dimensions. A Clear check shows that TQTTs can give a clear distinction of the topological type of 0- .1-, and 2-dimension manifolds (Andezej, 1980).
It can safely be concluded that The Classification of Two Dimensional Extended Topological Field Theories clearly gives a category that consists of morphisms and objects between the objects (Joel, 1994). The greater category is made up of two objects, morphisms between morphisms and morphisms between the two morphisms. However, there are several compositions of these morphisms, and other several coherent information associated with these many compositions.
Joel, S. Mathematical Quantum Theory I: Field Theory and many-body Theory, Dallas: AMS Bookstore, 1994
Andezej, P. Modern Trends in the Theory of Condensed Matter, New York: Springer-Verlag, 1980