A categorical view of algebraic theories

Abstract

[en] Classically, algebraic structures such as groups, rings, and many others were jointly studied with the language of universal algebra. It was later found that certain tools from category theory, called monads, are especially suitable to encode the whole amount of information contained in algebraic theories. In this work we discuss monads, and, in particular, some monads that are relevant in functional programming in Computer Science. We give a proof of the equivalence between the category of algebraic theories (formalized as Lawvere theories) and the category of finitary monads on the category of sets. We also prove that there is an equivalence between the category of algebras over a monad and the category of models of the associated Lawvere theory. Finally, we apply this equivalence of categories to give a new proof of the fact that all localizations on the category of abelian groups can be uniquely lifted to $R$-modules for every ring $R$.