The Stacks project

Definition 8.3.1. 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 of $\mathcal{C}$. Assume all the fibre products $U_ i \times _ U U_ j$, and $U_ i \times _ U U_ j \times _ U U_ k$ exist.

  1. A descent datum $(X_ i, \varphi _{ij})$ in $\mathcal{S}$ relative to the family $\{ f_ i : U_ i \to U\} $ is given by an object $X_ i$ of $\mathcal{S}_{U_ i}$ for each $i \in I$, an isomorphism $\varphi _{ij} : \text{pr}_0^*X_ i \to \text{pr}_1^*X_ j$ in $\mathcal{S}_{U_ i \times _ U U_ j}$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram

    \[ \xymatrix{ \text{pr}_0^*X_ i \ar[rd]_{\text{pr}_{01}^*\varphi _{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi _{ik}} & & \text{pr}_2^*X_ k \\ & \text{pr}_1^*X_ j \ar[ru]_{\text{pr}_{12}^*\varphi _{jk}} & } \]

    in the category $\mathcal{S}_{U_ i \times _ U U_ j \times _ U U_ k}$ commutes. This is called the cocycle condition.

  2. A morphism $\psi : (X_ i, \varphi _{ij}) \to (X'_ i, \varphi '_{ij})$ of descent data is given by a family $\psi = (\psi _ i)_{i\in I}$ of morphisms $\psi _ i : X_ i \to X'_ i$ in $\mathcal{S}_{U_ i}$ such that all the diagrams

    \[ \xymatrix{ \text{pr}_0^*X_ i \ar[r]_{\varphi _{ij}} \ar[d]_{\text{pr}_0^*\psi _ i} & \text{pr}_1^*X_ j \ar[d]^{\text{pr}_1^*\psi _ j} \\ \text{pr}_0^*X'_ i \ar[r]^{\varphi '_{ij}} & \text{pr}_1^*X'_ j \\ } \]

    in the categories $\mathcal{S}_{U_ i \times _ U U_ j}$ commute.

  3. The category of descent data relative to $\mathcal{U}$ is denoted $DD(\mathcal{U})$.

Comments (5)

Comment #7454 by Hao Peng on

I am confused by the cocycle condition. My worry is that the composition doesn't make sence: While is a morphism from to , is a morphism from to . Thus it is preassumed that , which is not in general true. Is this a mistake or we insect a natural isomorphism between them, or we can choose the cleavage such that this is true for any fiber products?

Comment #7606 by on

Dear Hao Peng, a computer couldn't parse this definition. However, we assume the human reader has read Section 8.2 where it is explained that we use the morphisms from Lemma 4.33.7 to identify these things. I do agree the text could be improved somewhat and I am happy to get explicit suggestions.

Comment #8881 by Siyuan Zheng on

is defined to be a map from to . However, in the diagram, the map from to is labeled as .

Comment #8882 by Siyuan Zheng on

Sorry, I realized the map in the diagram comes from the triple fiber product which is different from the map coming from the double fiber product.

There are also:

  • 4 comment(s) on Section 8.3: Descent data in fibred categories

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