Essentially Algebraic Descriptions of Locally Presentable Categories

Abstract

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 an 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 - or even wrong. Now the aim of the main part of this thesis is to give an intuitive proof of the mentioned fact, which covers existing examples, and can be generalized to the non-finitary case under mild assumptions. Here the set of sorts of such a description of a locally finitely presentable category is given by a strong generator of finitely presentables in this category. Also, this construction provides a new approach to the known characterization of quasivarieties. Further, the theory constructed for a locally finitely presentable category is some kind of clone. A non-trivial example of a locally finitely presentable category with a managable strong generator of finitely presentables is given by the category of coalgebras for a polynomial set-endofunctor. In the second part of this dissertation we show that this category is even equivalent to some variety of unary algebras without equations. Moreover, we characterize polynomial set-endofunctors by the property that the corresponding category of coalgebras is concretely equivalent to some presheaf category. Finally, we introduce for an endofunctor the concept of polynomiality w.r.t. a family of functors, and show that - under certain assumptions - several constructions and properties can be lifted to the corresponding category of coalgebras.