On the (algebraic) notion of 2-ring

TL;DR

This paper compares two definitions of 2-rings in category theory, clarifying their differences and the necessary conditions for their equivalence.

Contribution

It explains why an extra axiom is needed to unify the notions of Ann-category and categorical ring, using an equivalent description of symmetric monoidal categories.

Findings

Identifies the key difference between Ann-category and categorical ring notions.

Shows the necessity of an additional axiom for equivalence.

Provides an alternative characterization of symmetric monoidal categories.

Abstract

By a 2-ring we mean a groupoid with a structure analogous to that of a ring, up to coherent isomorphisms. Two different notions of 2-ring appear in the literature: the notion of {\em Ann-category}, due to Quang, and the notion of {\em categorical ring}, due to Jibladze and Pirashvili. The underlying data are the same in both cases, but the required axioms differ. In this note, we clarify the relationship between these notions by explaining why an additional axiom must be imposed for the two notions to be equivalent. Essential to this analysis is an equivalent description of a symmetric monoidal category.