Theory and Applications of Categories 11, 2003, 212-214
Publication year: 2003
Author’s retrospective remarks
The property of a category studied in the paper can be seen as a natural weakening of the property that all regular epimorphisms are normal epimorphisms. Pointed varieties having this latter property were called varieties with ideals by K. Fichtner, who obtained their syntactical characterization in 1968. Pointed regular categories having this property have been called normal categories in my paper in 2010. This term was independently suggested by my father, G. Janelidze. In universal algebra, there is a non-pointed version of this notion for which the term “0-regular variety” is used.
I showed in my PhD thesis that all product projections $A\times B\rightarrow B$ are normal if and only if the product projections $A\times A\rightarrow A$ are normal.
It is established in the paper that all pointed subtractive varieties and all Jónsson-Tarski varieties have normal projections. This means that pointed categories of essentially all structures of interest to algebra have normal projections. A counterexample if given by the category of pointed sets.
In my second talk at the Australian Category Seminar in 2003, where I showed that unital categories have normal projections. G. M. Kelly remarked during the talk that what now are called weakly unital categories also have normal projections. The notion of a weakly unital category is a N. Martins-Ferreira’s style of generalization of the notion of a unital category, and it was first studied in the PhD thesis of J. R. A. Gray.