  In this paper we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact
categories by axiomatising properties of $n\mathbb{Z}$-cluster tilting
subcategories. We study this 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 arise in this way, and discuss the problem for
$n\mathbb{Z}$-exact categories.
