A Diagrammatic Presentation of Equations in Categories

Abstract

Lifts of categorical diagrams D:𝖩→𝖷 against discrete opfibrations Ο€:𝖀→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≃Dβ€² 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 𝖷 along such equivalences is isomorphic to the localisation of the slice category 𝖒𝖺𝗍/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 𝖒𝖺𝗍.