Remark 48.32.4 (0G53)—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.

The Stacks project

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