The Stacks project

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

1. 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 $i \leq i' \leq i''$.

2. 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.