Nℤ-abelian and nℤ-exact categories

Abstract

Abstract In this paper, we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories by axiomatizing properties of $n\mathbb{Z}$-cluster tilting subcategories. We study these categories and show that every $n\mathbb{Z}$-cluster tilting subcategory of an abelian (resp., exact) category has a natural structure of an $n\mathbb{Z}$-abelian (resp., $n\mathbb{Z}$-exact) category. Also, we show that every small $n\mathbb{Z}$-abelian category arises in this way, and discuss the problem for $n\mathbb{Z}$-exact categories.