von Neumann regular category



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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 𝒮 𝒞 op\mathcal{S}^{\mathcal{C}^{op}} correspond to idempotent two-sided ideals JJ of the underlying small category 𝒞\mathcal{C}. What causes the problems for the general existence of Aufhebung at each level is the fact that an infinite intersection of such JJ is not necessarily idempotent itself. The condition occurring in the definition of a von Neumann regular category is a simple way to enforce this.


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


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


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


  • 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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