The inverse limit of $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}$ is the inverse limit of $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}$ in $\mathsf{Sets}$ as in 
, .
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1.
The Limit. The set $\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })$ defined by
\[ \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ (x_{\alpha })_{\alpha \in I}\in \prod _{\alpha \in I}X_{\alpha }\ \middle |\ \begin{aligned} & \text{for each $\alpha ,\beta \in I$, if $\alpha \preceq \beta $,}\\ & \text{then we have $x_{\alpha }=f_{\alpha \beta }(x_{\beta })$} \end{aligned} \right\} . \] -
2.
The Cone.The collection
\[ \left\{ \operatorname {\mathrm{\mathrm{pr}}}_{\gamma }\colon \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })\to X_{\gamma }\right\} _{\gamma \in I} \]of maps of sets defined as the restriction of the maps
\[ \left\{ \operatorname {\mathrm{\mathrm{pr}}}_{\gamma }\colon \prod _{\alpha \in I}X_{\alpha }\to X_{\gamma }\right\} _{\gamma \in I} \]of Item 2 of Construction 4.1.2.1.2 to $\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })$ and hence given by
\[ \operatorname {\mathrm{\mathrm{pr}}}_{\gamma }((x_{\alpha })_{\alpha \in I})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{\gamma } \]for each $\gamma \in I$ and each $(x_{\alpha })_{\alpha \in I}\in \displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })$.
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1.
The $p$-Adic Integers. The ring of $p$-adic integers $\mathbb {Z}_{p}$ of
is the inverse limit
\[ \mathbb {Z}_{p}\cong \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{n\in \mathbb {N}}(\mathbb {Z}_{/p^{n}}); \]see
.
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2.
Rings of Formal Power Series. The ring $R[\mspace {-3mu}[t]\mspace {-3mu}]$ of formal power series in a variable $t$ is the inverse limit
\[ R[\mspace {-3mu}[t]\mspace {-3mu}]\cong \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{n\in \mathbb {N}}(R[t]/t^{n}R[t]); \]see
.
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3.
Profinite Groups. Profinite groups are inverse limits of finite groups; see
.
4.1.6 Inverse Limits
Let $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}\colon (I,\preceq )\to \mathsf{Sets}$ be an inverse system of sets.
Concretely, the inverse limit of $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}$ is the pair $\smash {\Big(\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })}$, $\smash {\left\{ \operatorname {\mathrm{\mathrm{pr}}}_{\alpha }\right\} _{\alpha \in I}\Big)}$ consisting of:
Proof of Construction 4.1.6.1.2.
We claim that $\operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })$ is the limit of the inverse system of sets $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}$. First we need to check that the limit diagram defined by it commutes, i.e. that we have
\begin{align*} [f_{\alpha \beta }\circ \operatorname {\mathrm{\mathrm{pr}}}_{\alpha }]((x_{\gamma })_{\gamma \in I}) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{\alpha \beta }(\operatorname {\mathrm{\mathrm{pr}}}_{\alpha }((x_{\gamma })_{\gamma \in I}))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{\alpha \beta }(x_{\alpha })\\ & = x_{\beta }\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{\mathrm{pr}}}_{\beta }((x_{\gamma })_{\gamma \in I}), \end{align*}
where the third equality comes from the definition of $\operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })$. Next, we prove that $\operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })$ satisfies the universal property of an inverse limit. Suppose that we have, for each $\alpha ,\beta \in I$ with $\alpha \preceq \beta $, a diagram of the form
\[ \left\{ p_{\alpha }=\operatorname {\mathrm{\mathrm{pr}}}_{\alpha }\circ \phi \right\} _{\alpha \in I} \]
via
\[ \phi (\ell )=(p_{\alpha }(\ell ))_{\alpha \in I} \]
for each $\ell \in L$, where we note that $(p_{\alpha }(\ell ))_{\alpha \in I}\in \prod _{\alpha \in I}X_{\alpha }$ indeed lies in $\operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })$, as we have
\begin{align*} f_{\alpha \beta }(p_{\alpha }(\ell )) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[f_{\alpha \beta }\circ p_{\alpha }](\ell )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}p_{\beta }(\ell ) \end{align*}
for each $\beta \in I$ with $\alpha \preceq \beta $ by the commutativity of the diagram for $(L,\left\{ p_{\alpha }\right\} _{\alpha \in I})$.
Here are some examples of inverse limits of sets.
