On the categorical interpretation of ring cohomology

TL;DR

This paper investigates the axiomatic foundations of Ann-categories and categorical rings, proving their independence and establishing conditions under which they are equivalent, thereby clarifying their relationship in the context of ring cohomology.

Contribution

It proves the independence of Ann-category axioms and shows that adding an axiom to categorical rings yields an equivalent structure to Ann-categories.

Findings

Proved the independence of Ann-category axioms.

Established that adding an axiom makes categorical rings equivalent to Ann-categories.

Clarified the relationship between Ann-categories and categorical rings.

Abstract

In this paper, we have studied the axiomatics of {\it Ann-categories} and {\it categorical rings.} These are the categories with distributivity constraints whose axiomatics are similar with those of ring structures. The main result we have achieved is proving the independence of the axiomatics of Ann-category definition. And then we have proved that after adding an axiom into the definition of categorical rings, we obtain the new axiomatics which is equivalent to the one of Ann-categories. In [PJ], authors modified the definition of Ann-categories to be the one of {\it categorical rings,} where the condition Ann-1 is omitted, and the compatibility of the operation $\tx$ with the associativity and commutativity is replaced with the compatibility of the operation $\tx$ with the "associativity - commutativity" constraint. This replacement is to make it more convenient in using Mac Lane cohomology. However, the fact that the definition of categorical rings and Ann-categories are equivalent is still an open question. In [q], authors showed that the set of Ann-categories is a subset of the set of categorical rings. And these sets coincide if and only if the condition (U)is satisfied.