Definition 4.21.4. Let $I$ be a preordered set. We say a system (resp. inverse system) $(M_ i, f_{ii'})$ is a *directed system* (resp. *directed inverse system*) if $I$ is a directed set (Definition 4.21.1): $I$ is nonempty and for all $i_1, i_2 \in I$ there exists $i\in I$ such that $i_1 \leq i$ and $i_2 \leq i$.

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