Definition 15.2. Let $(I,\le )$ be a directed set and ${(G_i)}_{i\in I}$ a collection of groups together with maps ${\phi }_{ij}:G_j\to G_i$, $i\le j$ such that:

• ${\phi }_{ik}={\phi }_{ij}\circ {\phi }_{jk}$ for any $i\le j\le k$

• ${\phi }_{ii}=\mathrm{id}$

Say $({(G_i)}_{i=1},{\phi }_{ij})$ is an inverse system. The inverse limit of $(G_i,{\phi }_i)$ is

$$\underset{\stackrel{[}{i}]\leftarrow }{\lim }{G}_i=\{{(g_i)}_{i\in I}\in \prod _{i\in I}{G}_i|{\phi }_{ij}(g_j)=g_i\}.$$