A note on abelian envelopes

TL;DR

This paper clarifies the relationship between two concepts of universal abelian categories linked to exact categories, enabling easier conclusions and addressing some foundational questions in the field.

Contribution

It establishes a connection between Rump's quotient categories and Bodzenta-Bondal's abelian envelopes, providing new insights and answers to existing questions.

Findings

Connected two notions of universal abelian categories.

Derived several easy conclusions from the connection.

Answered basic questions raised by Bodzenta and Bondal.

Abstract

This is a short note bridging the gap between two notions of universal abelian categories associated to exact categories, namely, Rump's quotient categories and Bodzenta-Bondal's abelian envelopes. The established connection allows us to draw several easy conclusions, in particular, we answer some (baisc) questions raised by Bodzenta and Bondal.