On an Essentially Algebraic Theory for Locally Presentable Categories

Abstract

This paper presents a general construction, defining for each given strong generator ${\mathcal{G}}$ in any locally finitely presentable category ${\mathbf{C}}$ an essentially algebraic, finitary theory $\Gamma _{{\mathbf{C}}}$ – maximal in a certain sense – such that ${\mathbf{C}}$ is equivalent to the category of models ${\mathbf{Mod}}{\left( {\Gamma _{{\mathbf{C}}} } \right)}$ of $\Gamma _{{\mathbf{C}}}$. For regular generators ${\mathcal{G}}$, generalization to the non-finitary case is easily done, and yields a new proof of the famous characterization of many-sorted quasivarieties.