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category theory

topos theory

Contents

Idea

Von Neumann regular categories, or regular categories for short, are a class of small categories introduced by William Lawvere (2002) in the context of his theory of dimension and unity of opposites in order to provide presheaf toposes with well-behaved levels which in particular admit the Aufhebung of each level.

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)

References

• F. Borceux, J. Rosicky, On Von Neumann Varieties , TAC 13 no. 1 (2004) pp.5-26. (pdf)

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

• 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.

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