A categorical view of varieties and equations

We present a common framework to study varieties in great generality from a categorical point of view. The main application of this study is in the setting of algebraic categories, where we introduce Birkhoff varieties which are essentially subvarieties of algebraic categories, and we get a generalization of Birkhoff's variety theorem. In particular, we show that Birkhoff varieties are coreflexive equalizers. The key of this generalization is to give a more general concept of equation for subvarieties of algebraic categories. In order to get our characterization of Birkhoff varieties, we study inserters over algebraic categories, where we generalize some well-known results of algebras for finitary endofunctors over $Set$. By duality, we obtain a characterization of cosubvarieties of coalgebraic categories. Surprisingly, these cosubvarieties turn to be varieties according to our theory of varieties.