Item 2 (01EA) — The Clowder Project

Let $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}\colon (I,\preceq )\to \mathsf{Sets}$ be an inverse system of sets.

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 , .

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:

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 $(x_{\alpha })_{\alpha \in I}\in \displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })$.

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

for each $\alpha ,\beta \in I$ with $\alpha \preceq \beta $. Indeed, given $(x_{\gamma })_{\gamma \in I}\in \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\gamma \in I}(X_{\gamma })$, 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

in $\mathsf{Sets}$. Then there indeed exists a unique map $\phi \colon L\overset {\exists !}{\to }\smash {\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\operatorname {\mathrm{lim}}}}}}_{\alpha \in I}(X_{\alpha })}$ making the diagram

commute, being uniquely determined by the family of conditions

\[ \left\{ p_{\alpha }=\operatorname {\mathrm{\mathrm{pr}}}_{\alpha }\circ \phi \right\} _{\alpha \in I} \]

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.

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 .

4.1.6 Inverse Limits