The Stacks project

Definition 8.3.5. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice of pullbacks as in Categories, Definition 4.33.6. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ be a family of morphisms with target $U$. Assume all the fibre products $U_ i \times _ U U_ j$ and $U_ i \times _ U U_ j \times _ U U_ k$ exist.

  1. Given an object $X$ of $\mathcal{S}_ U$ the trivial descent datum is the descent datum $(X, \text{id}_ X)$ with respect to the family $\{ \text{id}_ U : U \to U\} $.

  2. Given an object $X$ of $\mathcal{S}_ U$ we have a canonical descent datum on the family of objects $f_ i^*X$ by pulling back the trivial descent datum $(X, \text{id}_ X)$ via the obvious map $\{ f_ i : U_ i \to U\} \to \{ \text{id}_ U : U \to U\} $. We denote this descent datum $(f_ i^*X, can)$.

  3. A descent datum $(X_ i, \varphi _{ij})$ relative to $\{ f_ i : U_ i \to U\} $ is called effective if there exists an object $X$ of $\mathcal{S}_ U$ such that $(X_ i, \varphi _{ij})$ is isomorphic to $(f_ i^*X, can)$.

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