Abstract
Lifts of categorical diagrams D:π©βπ· against discrete opfibrations Ο:π€βπ· 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 DβDβ² that precisely captures the idea of the two diagrams βhaving the same solutionsββ. We give such a definition, and then show how the localisation of the category of diagrams in π· along such equivalences is isomorphic to the localisation of the slice category π’πΊπ/π· 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 π’πΊπ.
