In this talk we describe representations of certain combinatorial categories of type \mathbb{N}^{\infty}, including the poset of positive integers with respect to division, the orbit category of the group (\mathbb{Z}, +), the category of finite acyclic groups and injective homomorphisms, and the opposite category of the category of finite acyclic groups and surjective homomorphisms, etc. We introduce a special norm on the object sets as well as certain shift functors,  and show that homological degrees of representations presented in finite degrees are bounded by their first two homological degrees. In particular, the category of representations presented in finite degrees is abelian.