In this paper, we investigate the property (P) that binary products commute with arbitrary coequalizers in pointed categories. Examples of such categories include any regular unital or (pointed) majority category with coequalizers, as well as any pointed factor permutable category with coequalizers. We establish a Mal'tsev term condition characterizing pointed varieties of universal algebras satisfying (P). We then consider categories satisfying (P) locally, i.e., those categories for which every fibre of the fibration of points satisfies (P). Examples include any regular Mal'tsev or majority category with coequalizers, as well as any regular Gumm category with coequalizers. Varieties satisfying (P) locally are also characterized by a Mal'tsev term condition, which turns out to be equivalent to a variant of Gumm's shifting lemma. Furthermore, we show that the varieties satisfying (P) locally are precisely the varieties with normal local projections in the sense of Z. Janelidze.
Keywords: product preserves coequalizers, stability of coequalizers under product, product commutes with coequalizers, normal projections, shifting lemma
2010 MSC: 08B05, 08B10, 18A30, 18B99, 03C05
Theory and Applications of Categories, Vol. 34, 2019, No. 43, pp 1386-1400.