nLab
abelian 2-category

Abelian categories are a common generalization of the category of abelian groups, categories of $R$-modules where $R$ is a ring, and the categories of sheaves of ${𝒪}_X$-modules where ${𝒪}_X$ is a sheaf of rings on a topological space $X$. Similarly, in the categorical dimension 2, one wants to define an abelian 2-category as a generalization of symmetric 2-groups and possibly some 2-related categories. There is a couple of different proposals in the literature.

• Mathieu Dupont, Abelian categories in dimension 2, arxiv/0809.1760
• Hiroyuki Nakaoka, Comparison of the definitions of Abelian 2-categories, arxiv/0904.0078