Inverse limit of a inverse system (algebraic objects)

We are going to deal with groups, but it probably works with other algebraic structures.

Think of a collection of groups ${A_{n}}_{n = 0}^{\infty}$ and morphisms:

θ_{n + 1} : A_{n + 1} \to A_{n}

We call this an inverse system. And we call a coherent sequence to a sequence $(a_{i})$ such that

θ_{i + 1} (a_{i + 1}) = a_{i}

The group of all these sequences is called the inverse limit and is usually denoted by

lim_{\leftarrow} A_{n}

[Atiyah 2018] page 103