Definition 4.21.2. Let $(I, \leq )$ be a preordered set. Let $\mathcal{C}$ be a category.

A

*system over $I$ in $\mathcal{C}$*, sometimes called a*inductive system over $I$ in $\mathcal{C}$*is given by objects $M_ i$ of $\mathcal{C}$ and for every $i \leq i'$ a morphism $f_{ii'} : M_ i \to M_{i'}$ such that $f_{ii} = \text{id}$ and such that $f_{ii''} = f_{i'i''} \circ f_{i i'}$ whenever $i \leq i' \leq i''$.An

*inverse system over $I$ in $\mathcal{C}$*, sometimes called a*projective system over $I$ in $\mathcal{C}$*is given by objects $M_ i$ of $\mathcal{C}$ and for every $i' \leq i$ a morphism $f_{ii'} : M_ i \to M_{i'}$ such that $f_{ii} = \text{id}$ and such that $f_{ii''} = f_{i'i''} \circ f_{i i'}$ whenever $i'' \leq i' \leq i$. (Note reversal of inequalities.)

We will say $(M_ i, f_{ii'})$ is a (inverse) system over $I$ to denote this. The maps $f_{ii'}$ are sometimes called the *transition maps*.

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