The Stacks project

Remark 4.21.3. Let $I$ be a preordered set. From $I$ we can construct a canonical partially ordered set $\overline{I}$ and an order preserving map $\pi : I \to \overline{I}$. Namely, we can define an equivalence relation $\sim $ on $I$ by the rule

\[ i \sim j \Leftrightarrow (i \leq j\text{ and }j \leq i). \]

We set $\overline{I} = I/\sim $ and we let $\pi : I \to \overline{I}$ be the quotient map. Finally, $\overline{I}$ comes with a unique partial ordering such that $\pi (i) \leq \pi (j) \Leftrightarrow i \leq j$. Observe that if $I$ is a directed set, then $\overline{I}$ is a directed partially ordered set. Given an (inverse) system $N$ over $\overline{I}$ we obtain an (inverse) system $M$ over $I$ by setting $M_ i = N_{\pi (i)}$. This construction defines a functor between the category of inverse systems over $I$ and $\overline{I}$. In fact, this is an equivalence. The reason is that if $i \sim j$, then for any system $M$ over $I$ the maps $M_ i \to M_ j$ and $M_ j \to M_ i$ are mutually inverse isomorphisms. More precisely, choosing a section $s : \overline{I} \to I$ of $\pi $ a quasi-inverse of the functor above sends $M$ to $N$ with $N_{\overline{i}} = M_{s(\overline{i})}$. Finally, this correspondence is compatible with colimits of systems: if $M$ and $N$ are related as above and if either $\mathop{\mathrm{colim}}\nolimits _{\overline{I}} N$ or $\mathop{\mathrm{colim}}\nolimits _ I M$ exists then so does the other and $\mathop{\mathrm{colim}}\nolimits _{\overline{I}} N = \mathop{\mathrm{colim}}\nolimits _ I M$. Similar results hold for inverse systems and limits of inverse systems.

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