To establish a set of axioms that describe or govern certain phenomena
1998, Jean-Pierre Marquis, “Book Review: Colin McLarty. Elementary Categories, Elementary Toposes.”, in Notre Dame Journal of Formal Logic, volume 39, number 3, page 441:
For toposes were invented by Grothendieck in the sixties for specific purposes in algebraic geometry and algebraic topology. Later on, Lawvere and Tierney  axiomatized the notion of an elementary topos with the goal of providing an appropriate foundation for differential geometry.