The successive dimension, without elegance

TL;DR

This paper investigates the structure of levels in toposes of presheaves, introducing a generalized successor function applicable to broader categories beyond Reedy elegant ones, revealing new insights into their hierarchical organization.

Contribution

It extends the concept of the successor function to presheaf toposes on categories with split-epi/mono factorizations, generalizing previous results beyond Reedy elegant categories.

Findings

The successor function can be defined on the poset of full subcategories closed under subobjects.

This generalized successor differs from Lawvere's Aufhebung in certain cases.

The approach applies to categories like skeletal, graphic, or regular categories.

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.