Let $\{ G_i \}_{i=0}^{\infty}$ be a sequence of groups which are related by a chain of surjective homomorphisms $f_i\colon G_i \to G_{i-1}$ such that

Note: The inverse limit of $G_i$ can be checked to be a subgroup of the product $\prod_{i=0}^{\infty} G_i$ . See below for a more general definition.

Examples:

1. Let $p\in\Nats$ be a prime. Let $G_0=\{0\}$ and $G_i=\Ints/p^i\Ints$ . Define the connecting homomorphisms $f_i$ , for $i\geq 2$ , to be ``reduction modulo $p^{i-1}$ '' i.e. $$f_i\colon \Ints/p^i\Ints \to \Ints/p^{i-1}\Ints$$ $$f_i(x \operatorname{mod}p^i)= x \operatorname{mod} p^{i-1}$$ which are obviously surjective homomorphisms. The inverse limit of $(\Ints/p^i\Ints, f_i)$ is called the $p$ -adic integers and denoted by $$\Ints_p=\varprojlim\Ints/p^i\Ints$$
2. Let $E$ be an elliptic curve defined over $\Complex$ . Let $p$ be a prime and for any natural number $n$ write $E[n]$ for the $n$ -torsion group, i.e. $$E[n]=\{ Q\in E \mid n\cdot Q=O\}$$ In this case we define $G_i=E[p^i]$ , and $$f_i\colon E[p^i] \to E[p^{i-1}], \quad f_i(Q)=p\cdot Q$$ The inverse limit of $(E[p^i],f_i)$ is called the Tate module of $E$ and denoted $$T_p(E)=\varprojlim E[p^i]$$

The concept of inverse limit can be defined in far more generality. Let $(S,\leq)$ be a directed set and let $\mathcal{C}$ be a category. Let $\{ G_{\alpha}\}_{\alpha\in S}$ be a collection of objects in the category $\mathcal{C}$ and let $$\{f_{\alpha,\beta}\colon G_{\beta} \to G_{\alpha}\mid \alpha,\beta \in S,\quad \alpha\leq\beta \}$$ be a collection of morphisms satisfying:

1. For all $\alpha\in S$ , $f_{\alpha,\alpha}=\operatorname{Id}_{G_{\alpha}}$ , the identity morphism.
2. For all $\alpha,\beta,\gamma\in S$ such that $\alpha \leq \beta \leq \gamma$ , we have $f_{\alpha,\gamma} = f_{\alpha,\beta} \circ f_{\beta,\gamma}$ (composition of morphisms).

For a good example of this more general construction, see infinite Galois theory.