The diagrammatic presentation of equations in categories

Abstract

Lifts of categorical diagrams $D\colon\mathsf{J}\to\mathsf{X}$ against discrete opfibrations $\pi\colon\mathsf{E}\to\mathsf{X}$ can be interpreted as presenting solutions to systems of equations. With this interpretation in mind, it is natural to ask if there is a notion of equivalence of diagrams $D\simeq D'$ that precisely captures the idea of the two diagrams "having the same solutions''. We give such a definition, and then show how the localisation of the category of diagrams in $\mathsf{X}$ along such equivalences is isomorphic to the localisation of the slice category $\mathsf{Cat}/\mathsf{X}$ along the class of initial functors. Finally, we extend this result to the 2-categorical setting, proving the analogous statement for any locally presentable 2-category in place of $\mathsf{Cat}$.