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