The Stacks project

Remark 48.32.4. Let $\mathcal{C}$ be a category. Suppose given an inverse system

\[ \ldots \xrightarrow {\alpha _4} (M_{3, n}) \xrightarrow {\alpha _3} (M_{2, n}) \xrightarrow {\alpha _2} (M_{1, n}) \]

of inverse systems in the category of pro-objects of $\mathcal{C}$. In other words, the arrows $\alpha _ i$ are morphisms of pro-objects. By Categories, Example 4.22.6 we can represent each $\alpha _ i$ by a pair $(m_ i, a_ i)$ where $m_ i : \mathbf{N} \to \mathbf{N}$ is an increasing function and $a_{i, n} : M_{i, m_ i(n)} \to M_{i - 1, n}$ is a morphism of $\mathcal{C}$ making the diagrams

\[ \xymatrix{ \ldots \ar[r] & M_{i, m_ i(3)} \ar[d]^{a_{i, 3}} \ar[r] & M_{i, m_ i(2)} \ar[d]^{a_{i, 2}} \ar[r] & M_{i, m_ i(1)} \ar[d]^{a_{i, 1}} \\ \ldots \ar[r] & M_{i - 1, 3} \ar[r] & M_{i - 1, 2} \ar[r] & M_{i - 1, 1} } \]

commute. By replacing $m_ i(n)$ by $\max (n, m_ i(n))$ and adjusting the morphisms $a_ i(n)$ accordingly (as in the example referenced) we may assume that $m_ i(n) \geq n$. In this situation consider the inverse system

\[ \ldots \to M_{4, m_4(m_3(m_2(4)))} \to M_{3, m_3(m_2(3))} \to M_{2, m_2(2)} \to M_{1, 1} \]

with general term

\[ M_ k = M_{k, m_ k(m_{k - 1}(\ldots (m_2(k))\ldots ))} \]

For any object $N$ of $\mathcal{C}$ we have

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_{i, n}, N) = \mathop{\mathrm{colim}}\nolimits _ k \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ k, N) \]

We omit the details. In other words, we see that the inverse system $(M_ k)$ has the property

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _{\text{Pro-}\mathcal{C}}((M_{i, n}), N) = \mathop{\mathrm{Mor}}\nolimits _{\text{Pro-}\mathcal{C}}((M_ k), N) \]

This property determines the inverse system $(M_ k)$ up to pro-isomorphism by the discussion in Categories, Remark 4.22.7. In this way we can turn certain inverse systems in $\text{Pro-}\mathcal{C}$ into pro-objects with countable index categories.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0G53. Beware of the difference between the letter 'O' and the digit '0'.