Products and coequalizers in pointed categories

TL;DR

This paper characterizes categories where finite products commute with coequalizers, providing Mal'tsev conditions and exploring local versions, with implications for various algebraic and categorical structures.

Contribution

It introduces a Mal'tsev term condition characterizing categories satisfying property (P) and its local variants, linking them to normal local projections.

Findings

Categories with property (P) include regular unital and majority categories.

Varieties satisfying (P) locally are characterized by a Mal'tsev term condition.

Such varieties are exactly those with normal local projections.

Abstract

In this paper, we investigate the property (P) that finite 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 $Pt_{\mathbb{C}}(X)$ of the fibration of points $\pi: Pt_{\mathbb{C}} \rightarrow \mathbb{C}$ 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.