As shown in (Kelly-Lawvere 1989) essential subtoposes of presheaf toposes $\mathcal{S}^{\mathcal{C}^{op}}$ correspond to idempotent two-sided ideals $J$ of the underlying small category $\mathcal{C}$. What causes the problems for the general existence of Aufhebung at each level is the fact that an infinite intersection of such $J$ is not necessarily idempotent itself. The condition occurring in the definition of a von Neumann regular category is a simple way to enforce this.

Definition

A small category $\mathcal{C}$ is called von Neumann regular if all two-sided ideals are idempotent.

Remark

The terminology is presumably chosen in view of the concept of a von Neumann regular ring$R$ i.e. one such that every $a \in R$ has a ‘weak inverse’ $\bar{a}$ with $a = a \bar{a} a$ as the following property illustrates.

Properties

$\mathcal{C}$ is von Neumann regular iff for any morphism $a$ in $\mathcal{C}$ there exists a reverse morphism $\bar{a}$ and two endomorphisms $x,y$ with $a=y a \bar{a} a x$. (Lawvere 2002)

G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations , Bull.Soc.Math. de Belgique XLI (1989) 261-299 [pdf]

F. W. Lawvere, Display of graphics and their applications, as exemplified by 2-categories and the Hegelian “taco” , Proceedings of the first international conference on algebraic methodology and software technology University of Ioowa, May 22-24 1989, Iowa City, pp.51-74.

F. W. Lawvere, Linearization of graphic toposes via Coxeter groups , JPAA 168 (2002) pp.425-436.

Last revised on March 27, 2023 at 07:21:45.
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