Essentially Algebraic Descriptions of Locally Presentable Categories

Local presentability has turned out to be one of the most fruitful concepts in category theory. The fact, that a category is locally finitely presentable iff it is equivalent to the category of models of some essentially algebraic, finitary theory, is widely known. Unfortunately, the existing approaches in literature are either unsatisfactory - with respect to existing examples and to the number of sorts needed - or even wrong. This thesis provides an intuitive proof of the mentioned fact, which covers existing examples, and can be generalized to the non-finitary case under mild assumptions. In addition to that it presents a new approach to the known characterization of quasivarieties. A non-trivial example is given by the category of coalgebras for a polynomial set-endofunctor, which turns out to be equivalent to some variety of unary algebras without equations. Moreover, polynomial set-endofunctors are characterized by the property that the corresponding category of coalgebras is concretely equivalent to some presheaf category.