Equivalences between certain algebraic categories

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful. It creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. For instance using categorical equivalences one can transform topological problems to algebraic problems. In this thesis it has been shown that it is enough to have a full, faithfull and dense functor to show the equivalence of the categories, and this result is explained by many examples. We then recalled the concept of p-categorical equivalence for algebraic structures and obtained some conclusions about the p-categorical equivalence. Finally, the relationship between a categorical equivalence and unitary adjoint functors has been shown.