The Stacks project

Lemma 8.3.7. Let $\mathcal{C}$ be a category. Let $\mathcal{V} = \{ V_ j \to U\} _{j \in J} \to \mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a morphism of families of maps with fixed target of $\mathcal{C}$ given by $\text{id} : U \to U$, $\alpha : J \to I$ and $f_ j : V_ j \to U_{\alpha (j)}$. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. If

  1. for $0 \leq p \leq 3$ and $0 \leq q \leq 3$ with $p + q \geq 2$ and $i_1, \ldots , i_ p \in I$ and $j_1, \ldots , j_ q \in J$ the fibre products $U_{i_1} \times _ U \ldots \times _ U U_{i_ p} \times _ U V_{j_1} \times _ U \ldots \times _ U V_{j_ q}$ exist,

  2. the functor $\mathcal{S}_ U \to DD(\mathcal{V})$ is an equivalence,

  3. for every $i \in I$ the functor $\mathcal{S}_{U_ i} \to DD(\mathcal{V}_ i)$ is fully faithful, and

  4. for every $i, i' \in I$ the functor $\mathcal{S}_{U_ i \times _ U U_{i'}} \to DD(\mathcal{V}_{ii'})$ is faithful.

Here $\mathcal{V}_ i = \{ U_ i \times _ U V_ j \to U_ i\} _{j \in J}$ and $\mathcal{V}_{ii'} = \{ U_ i \times _ U U_{i'} \times _ U V_ j \to U_ i \times _ U U_{i'}\} _{j \in J}$. Then $\mathcal{S}_ U \to DD(\mathcal{U})$ is an equivalence.

Proof. Condition (1) guarantees we have enough fibre products so that the statement makes sense. We will show that the functor $\mathcal{S}_ U \to DD(\mathcal{U})$ is essentially surjective. Suppose given a descent datum $(X_ i, \varphi _{ii'})$ relative to $\mathcal{U}$. By Lemma 8.3.3 we can pull this back to a descent datum $(X_ j, \varphi _{jj'})$ for $\mathcal{V}$. By assumption (2) this descent datum is effective, hence we get an object $X$ of $\mathcal{S}_ U$ such that the pullback of the trivial descent datum $(X, \text{id}_ X)$ by the morphism $\mathcal{V} \to \{ U \to U\} $ is isomorphic to $(X_ j, \varphi _{jj'})$. Next, observe that we have a diagram

\[ \xymatrix{ \mathcal{V}_ i \ar[r] \ar[d] & \mathcal{V} \ar[r] & \mathcal{U} \ar[d] \\ \{ U_ i \to U_ i\} \ar[rr] \ar[rru] & & \{ U \to U\} } \]

of morphisms of families of maps with fixed target of $\mathcal{C}$. This diagram does not commute, but by Lemma 8.3.3 the pullback functors on descent data one gets are canonically isomorphic. Hence $(X, \text{id}_ X)$ and $(X_ i, \text{id}_{X_ i})$ pull back to isomorphic objects in $DD(\mathcal{V}_ i)$. Hence by assumption (3) we obtain an isomorphism $(U_ i \to U)^*X \to X_ i$ in the category $\mathcal{S}_{U_ i}$. We omit the verification that these arrows are compatible with the morphisms $\varphi _{ii'}$; hint: use the faithfulness of the functors in condition (4). We also omit the verification that the functor $\mathcal{S}_ U \to DD(\mathcal{U})$ is fully faithful. $\square$

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