The successive dimension, without elegance

Abstract

Experience shows that the poset of levels (or dimensions) of the topos of presheaves on some elegant Reedy categories may be equipped with a monotone increasing `successor' function which, as the case of simplicial sets shows, is different from Lawvere's Aufhebung in general. We prove that a similar result holds for the topos of presheaves on a small category with split-epi/mono factorizations; a typical feature of categories that are Reedy elegant, or skeletal, or graphic (von Neumann-)regular, but more general. In fact, we show that the more general `successor' may be described as a function on the poset of full subcategories of the site that are closed under subobjects.