## 8.3 Descent data in fibred categories

In this section we define the notion of a descent datum in the abstract setting of a fibred category. Before we do so we point out that this is completely analogous to descent data for quasi-coherent sheaves (Descent, Section 35.2) and descent data for schemes over schemes (Descent, Section 35.34).

We will use the convention where the projection maps $\text{pr}_ i : X \times \ldots \times X \to X$ are labeled starting with $i = 0$. Hence we have $\text{pr}_0, \text{pr}_1 : X \times X \to X$, $\text{pr}_0, \text{pr}_1, \text{pr}_2 : X \times X \times X \to X$, etc.

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 $DD(\mathcal{U})$.

The fibre products $U_ i \times _ U U_ j$ and $U_ i \times _ U U_ j \times _ U U_ k$ will exist if each of the morphisms $f_ i : U_ i \to U$ is representable, see Categories, Definition 4.6.4. Recall that in a site one of the conditions for a covering $\{ U_ i \to U\}$ is that each of the morphisms is representable, see Sites, Definition 7.6.2 part (3). In fact the main interest in the definition above is where $\mathcal{C}$ is a site and $\{ U_ i \to U\}$ is a covering of $\mathcal{C}$. However, a descent datum is just an abstract gadget that can be defined as above. This is useful: for example, given a fibred category over $\mathcal{C}$ one can look at the collection of families with respect to which descent data are effective, and try to use these as the family of coverings for a site.

Remarks 8.3.2. Two remarks on Definition 8.3.1 are in order. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Let $\{ f_ i : U_ i \to U\} _{i \in I}$, and $(X_ i, \varphi _{ij})$ be as in Definition 8.3.1.

1. There is a diagonal morphism $\Delta : U_ i \to U_ i \times _ U U_ i$. We can pull back $\varphi _{ii}$ via this morphism to get an automorphism $\Delta ^\ast \varphi _{ii} \in \text{Aut}_{U_ i}(X_ i)$. On pulling back the cocycle condition for the triple $(i, i, i)$ by $\Delta _{123} : U_ i \to U_ i \times _ U U_ i \times _ U U_ i$ we deduce that $\Delta ^\ast \varphi _{ii} \circ \Delta ^\ast \varphi _{ii} = \Delta ^\ast \varphi _{ii}$; thus $\Delta ^\ast \varphi _{ii} = \text{id}_{X_ i}$.

2. There is a morphism $\Delta _{13}: U_ i \times _ U U_ j \to U_ i \times _ U U_ j \times _ U U_ i$ and we can pull back the cocycle condition for the triple $(i, j, i)$ to get the identity $(\sigma ^\ast \varphi _{ji}) \circ \varphi _{ij} = \text{id}_{\text{pr}_0^\ast X_ i}$, where $\sigma : U_ i \times _ U U_ j \to U_ j \times _ U U_ i$ is the switching morphism.

Lemma 8.3.3. (Pullback of descent data.) Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice pullbacks as in Categories, Definition 4.33.6. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$, and $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ be a families of morphisms of $\mathcal{C}$ with fixed target. Assume all the fibre products $U_ i \times _ U U_{i'}$, $U_ i \times _ U U_{i'} \times _ U U_{i''}$, $V_ j \times _ V V_{j'}$, and $V_ j \times _ V V_{j'} \times _ V V_{j''}$ exist. Let $\alpha : I \to J$, $h : U \to V$ and $g_ i : U_ i \to V_{\alpha (i)}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1.

1. Let $(Y_ j, \varphi _{jj'})$ be a descent datum relative to the family $\{ V_ j \to V\}$. The system

$\left( g_ i^*Y_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')} \right)$

is a descent datum relative to $\mathcal{U}$.

2. This construction defines a functor between descent data relative to $\mathcal{V}$ and descent data relative to $\mathcal{U}$.

3. Given a second $\alpha ' : I \to J$, $h' : U \to V$ and $g'_ i : U_ i \to V_{\alpha '(i)}$ morphism of families of maps with fixed target, then if $h = h'$ the two resulting functors between descent data are canonically isomorphic.

Proof. Omitted. $\square$

Definition 8.3.4. With $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$, $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$, $\alpha : I \to J$, $h : U \to V$, and $g_ i : U_ i \to V_{\alpha (i)}$ as in Lemma 8.3.3 the functor

$(Y_ j, \varphi _{jj'}) \longmapsto (g_ i^*Y_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')})$

constructed in that lemma is called the pullback functor on descent data.

Given $h : U \to V$, if there exists a morphism $\tilde h : \mathcal{U} \to \mathcal{V}$ covering $h$ then $\tilde h^*$ is independent of the choice of $\tilde h$ as we saw in Lemma 8.3.3. Hence we will sometimes simply write $h^*$ to indicate the pullback functor.

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 $(f_ i^*X, can)$.

Note that the rule that associates to $X \in \mathcal{S}_ U$ its canonical descent datum relative to $\mathcal{U}$ defines a functor

$\mathcal{S}_ U \longrightarrow DD(\mathcal{U}).$

A descent datum is effective if and only if it is in the essential image of this functor. Let us make explicit the canonical descent datum as follows.

Lemma 8.3.6. In the situation of Definition 8.3.5 part (2) the maps $can_{ij} : \text{pr}_0^*f_ i^*X \to \text{pr}_1^*f_ j^*X$ are equal to $(\alpha _{\text{pr}_1, f_ j})_ X \circ (\alpha _{\text{pr}_0, f_ i})_ X^{-1}$ where $\alpha _{\cdot , \cdot }$ is as in Categories, Lemma 4.33.7 and where we use the equality $f_ i \circ \text{pr}_0 = f_ j \circ \text{pr}_1$ as maps $U_ i \times _ U U_ j \to U$.

Proof. Omitted. $\square$

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. $\square$

Comment #5502 by Quang Dao on

I am having trouble proving (3) of Lemma 02ZD. Could you give a hint as to how the isomorphism is constructed?

Comment #5703 by on

The hint is to pullback the isomorphisms $\varphi_{\alpha(i)\alpha'(i)}$ by the morphisms $(g_i, g'_i) : U_i \to V_{\alpha(i)} \times_V V_{\alpha'(i)}$ to get the isomorphism between pullbacks of descent data. This is similar to the fact that the map on Cech cohomology doesn't depend on which map of coverings you choose.

I think the $x_i$ in Remark 026C should be capital $X_i$.

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