The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

8.4 Stacks

Here is the definition of a stack. It mixes the notion of a fibred category with the notion of descent.

Definition 8.4.1. Let $\mathcal{C}$ be a site. A stack over $\mathcal{C}$ is a category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$ which satisfies the following conditions:

  1. $p : \mathcal{S} \to \mathcal{C}$ is a fibred category, see Categories, Definition 4.32.5,

  2. for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any $x, y \in \mathcal{S}_ U$ the presheaf $\mathit{Mor}(x, y)$ (see Definition 8.2.2) is a sheaf on the site $\mathcal{C}/U$, and

  3. for any covering $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ of the site $\mathcal{C}$, any descent datum in $\mathcal{S}$ relative to $\mathcal{U}$ is effective.

We find the formulation above the most convenient way to think about a stack. Namely, given a category over $\mathcal{C}$ in order to verify that it is a stack you proceed to check properties (1), (2) and (3) in that order. Certainly properties (2) and (3) do not make sense if the category isn't fibred. Without (2) we cannot prove that the descent in (3) is unique up to unique isomorphism and functorial.

The following lemma provides an alternative definition.

Lemma 8.4.2. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category over $\mathcal{C}$. The following are equivalent

  1. $\mathcal{S}$ is a stack over $\mathcal{C}$, and

  2. for any covering $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ of the site $\mathcal{C}$ the functor

    \[ \mathcal{S}_ U \longrightarrow DD(\mathcal{U}) \]

    which associates to an object its canonical descent datum is an equivalence.

Proof. Omitted. $\square$

Lemma 8.4.3. Let $p : \mathcal{S} \to \mathcal{C}$ be a stack over the site $\mathcal{C}$. Let $\mathcal{S}'$ be a subcategory of $\mathcal{S}$. Assume

  1. if $\varphi : y \to x$ is a strongly cartesian morphism of $\mathcal{S}$ and $x$ is an object of $\mathcal{S}'$, then $y$ is isomorphic to an object of $\mathcal{S}'$,

  2. $\mathcal{S}'$ is a full subcategory of $\mathcal{S}$, and

  3. if $\{ f_ i : U_ i \to U\} $ is a covering of $\mathcal{C}$, and $x$ an object of $\mathcal{S}$ over $U$ such that $f_ i^*x$ is isomorphic to an object of $\mathcal{S}'$ for each $i$, then $x$ is isomorphic to an object of $\mathcal{S}'$.

Then $\mathcal{S}' \to \mathcal{C}$ is a stack.

Proof. Omitted. Hints: The first condition guarantees that $\mathcal{S}'$ is a fibred category. The second condition guarantees that the $\mathit{Isom}$-presheaves of $\mathcal{S}'$ are sheaves (as they are identical to their counter parts in $\mathcal{S}$). The third condition guarantees that the descent condition holds in $\mathcal{S}'$ as we can first descend in $\mathcal{S}$ and then (3) implies the resulting object is isomorphic to an object of $\mathcal{S}'$. $\square$

Lemma 8.4.4. Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be categories over $\mathcal{C}$. Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$. Then $\mathcal{S}_1$ is a stack over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is a stack over $\mathcal{C}$.

Proof. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$, $G : \mathcal{S}_2 \to \mathcal{S}_1$ be functors over $\mathcal{C}$, and let $i : F \circ G \to \text{id}_{\mathcal{S}_2}$, $j : G \circ F \to \text{id}_{\mathcal{S}_1}$ be isomorphisms of functors over $\mathcal{C}$. By Categories, Lemma 4.32.8 we see that $\mathcal{S}_1$ is fibred if and only if $\mathcal{S}_2$ is fibred over $\mathcal{C}$. Hence we may assume that both $\mathcal{S}_1$ and $\mathcal{S}_2$ are fibred. Moreover, the proof of Categories, Lemma 4.32.8 shows that $F$ and $G$ map strongly cartesian morphisms to strongly cartesian morphisms, i.e., $F$ and $G$ are $1$-morphisms of fibred categories over $\mathcal{C}$. This means that given $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $x, y \in \mathcal{S}_{1, U}$ then the presheaves

\[ \mathit{Mor}_{\mathcal{S}_1}(x, y), \mathit{Mor}_{\mathcal{S}_1}(F(x), F(y)) : (\mathcal{C}/U)^{opp} \longrightarrow \textit{Sets}. \]

are identified, see Lemma 8.2.3. Hence the first is a sheaf if and only if the second is a sheaf. Finally, we have to show that if every descent datum in $\mathcal{S}_1$ is effective, then so is every descent datum in $\mathcal{S}_2$. To do this, let $(X_ i, \varphi _{ii'})$ be a descent datum in $\mathcal{S}_2$ relative the covering $\{ U_ i \to U\} $ of the site $\mathcal{C}$. Then $(G(X_ i), G(\varphi _{ii'}))$ is a descent datum in $\mathcal{S}_1$ relative the covering $\{ U_ i \to U\} $. Let $X$ be an object of $\mathcal{S}_{1, U}$ such that the descent datum $(f_ i^*X, can)$ is isomorphic to $(G(X_ i), G(\varphi _{ii'}))$. Then $F(X)$ is an object of $\mathcal{S}_{2, U}$ such that the descent datum $(f_ i^*F(X), can)$ is isomorphic to $(F(G(X_ i)), F(G(\varphi _{ii'})))$ which in turn is isomorphic to the original descent datum $(X_ i, \varphi _{ii'})$ using $i$. $\square$

The $2$-category of stacks over $\mathcal{C}$ is defined as follows.

Definition 8.4.5. Let $\mathcal{C}$ be a site. The $2$-category of stacks over $\mathcal{C}$ is the sub $2$-category of the $2$-category of fibred categories over $\mathcal{C}$ (see Categories, Definition 4.32.9) defined as follows:

  1. Its objects will be stacks $p : \mathcal{S} \to \mathcal{C}$.

  2. Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$ and such that $G$ maps strongly cartesian morphisms to strongly cartesian morphisms.

  3. Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_ x) = \text{id}_{p(x)}$ for all $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$.

Lemma 8.4.6. Let $\mathcal{C}$ be a site. The $(2, 1)$-category of stacks over $\mathcal{C}$ has 2-fibre products, and they are described as in Categories, Lemma 4.31.3.

Proof. Let $f : \mathcal{X} \to \mathcal{S}$ and $g : \mathcal{Y} \to \mathcal{S}$ be $1$-morphisms of stacks over $\mathcal{C}$ as defined above. The category $\mathcal{X} \times _\mathcal {S} \mathcal{Y}$ described in Categories, Lemma 4.31.3 is a fibred category according to Categories, Lemma 4.32.10. (This is where we use that $f$ and $g$ preserve strongly cartesian morphisms.) It remains to show that the morphism presheaves are sheaves and that descent relative to coverings of $\mathcal{C}$ is effective.

Recall that an object of $\mathcal{X} \times _\mathcal {S} \mathcal{Y}$ is given by a quadruple $(U, x, y, \phi )$. It lies over the object $U$ of $\mathcal{C}$. Next, let $(U, x', y', \phi ')$ be second object lying over $U$. Recall that $\phi : f(x) \to g(y)$, and $\phi ' : f(x') \to g(y')$ are isomorphisms in the category $\mathcal{S}_ U$. Let us use these isomorphisms to identify $z = f(x) = g(y)$ and $z' = f(x') = g(y')$. With this identifications it is clear that

\[ \mathit{Mor}((U, x, y, \phi ), (U, x', y', \phi ')) = \mathit{Mor}(x, x') \times _{\mathit{Mor}(z, z')} \mathit{Mor}(y, y') \]

as presheaves. However, as the fibred product in the category of presheaves preserves sheaves (Sites, Lemma 7.10.1) we see that this is a sheaf.

Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ be a covering of the site $\mathcal{C}$. Let $(X_ i, \chi _{ij})$ be a descent datum in $\mathcal{X} \times _\mathcal {S} \mathcal{Y}$ relative to $\mathcal{U}$. Write $X_ i = (U_ i, x_ i, y_ i, \phi _ i)$ as above. Write $\chi _{ij} = (\varphi _{ij}, \psi _{ij})$ as in the definition of the category $\mathcal{X} \times _\mathcal {S} \mathcal{Y}$ (see Categories, Lemma 4.31.3). It is clear that $(x_ i, \varphi _{ij})$ is a descent datum in $\mathcal{X}$ and that $(y_ i, \psi _{ij})$ is a descent datum in $\mathcal{Y}$. Since $\mathcal{X}$ and $\mathcal{Y}$ are stacks these descent data are effective. Thus we get $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)$, and $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$ with $x_ i = x|_{U_ i}$, and $y_ i = y|_{U_ i}$ compatibly with descent data. Set $z = f(x)$ and $z' = g(y)$ which are both objects of $\mathcal{S}_ U$. The morphisms $\phi _ i$ are elements of $\mathit{Isom}(z, z')(U_ i)$ with the property that $\phi _ i|_{U_ i \times _ U U_ j} = \phi _ j|_{U_ i \times _ U U_ j}$. Hence by the sheaf property of $\mathit{Isom}(z, z')$ we obtain an isomorphism $\phi : z = f(x) \to z' = g(y)$. We omit the verification that the canonical descent datum associated to the object $(U, x, y, \phi )$ of $(\mathcal{X} \times _\mathcal {S} \mathcal{Y})_ U$ is isomorphic to the descent datum we started with. $\square$

Lemma 8.4.7. Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be stacks over $\mathcal{C}$. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$ be a $1$-morphism. Then the following are equivalent

  1. $F$ is fully faithful,

  2. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and for every $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{1, U})$ the map

    \[ F : \mathit{Mor}_{\mathcal{S}_1}(x, y) \longrightarrow \mathit{Mor}_{\mathcal{S}_2}(F(x), F(y)) \]

    is an isomorphism of sheaves on $\mathcal{C}/U$.

Proof. Assume (1). For $U, x, y$ as in (2) the displayed map $F$ evaluates to the map $F : \mathop{Mor}\nolimits _{\mathcal{S}_{1, V}}(x|_ V, y|_ V) \to \mathop{Mor}\nolimits _{\mathcal{S}_{2, V}}(F(x|_ V), F(y|_ V))$ on an object $V$ of $\mathcal{C}$ lying over $U$. Now, since $F$ is fully faithful, the corresponding map $\mathop{Mor}\nolimits _{\mathcal{S}_1}(x|_ V, y|_ V) \to \mathop{Mor}\nolimits _{\mathcal{S}_2}(F(x|_ V), F(y|_ V))$ is a bijection. Morphisms in the fibre category $\mathcal{S}_{1, V}$ are exactly those morphisms between $x|_ V$ and $y|_ V$ in $\mathcal{S}_1$ lying over $\text{id}_ V$. Similarly, morphisms in the fibre category $\mathcal{S}_{2, V}$ are exactly those morphisms between $F(x|_ V)$ and $F(y|_ V)$ in $\mathcal{S}_2$ lying over $\text{id}_ V$. Thus we find that $F$ induces a bijection between these also. Hence (2) holds.

Assume (2). Suppose given objects $U$, $V$ of $\mathcal{C}$ and $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{1, U})$ and $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{1, V})$. To show that $F$ is fully faithful, it suffices to prove it induces a bijection on morphisms lying over a fixed $f : U \to V$. Choose a strongly Cartesian $f^*y \to y$ in $\mathcal{S}_1$ lying above $f$. This results in a bijection between the set of morphisms $x \to y$ in $\mathcal{S}_1$ lying over $f$ and $\mathop{Mor}\nolimits _{\mathcal{S}_{1, U}}(x, f^*y)$. Since $F$ preserves strongly Cartesian morphisms as a $1$-morphism in the $2$-category of stacks over $\mathcal{C}$, we also get a bijection between the set of morphisms $F(x) \to F(y)$ in $\mathcal{S}_2$ lying over $f$ and $\mathop{Mor}\nolimits _{\mathcal{S}_{2, U}}(F(x), F(f^*y))$. Since $F$ induces a bijection $\mathop{Mor}\nolimits _{\mathcal{S}_{1, U}}(x, f^*y) \to \mathop{Mor}\nolimits _{\mathcal{S}_{2, U}}(F(x), F(f^*y))$ we conclude (1) holds. $\square$

Lemma 8.4.8. Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be stacks over $\mathcal{C}$. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$ be a $1$-morphism which is fully faithful. Then the following are equivalent

  1. $F$ is an equivalence,

  2. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and for every $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{2, U})$ there exists a covering $\{ f_ i : U_ i \to U\} $ such that $f_ i^*x$ is in the essential image of the functor $F : \mathcal{S}_{1, U_ i} \to \mathcal{S}_{2, U_ i}$.

Proof. The implication (1) $\Rightarrow $ (2) is immediate. To see that (2) implies (1) we have to show that every $x$ as in (2) is in the essential image of the functor $F$. To do this choose a covering as in (2), $x_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{1, U_ i})$, and isomorphisms $\varphi _ i : F(x_ i) \to f_ i^*x$. Then we get a descent datum for $\mathcal{S}_1$ relative to $\{ f_ i : U_ i \to U\} $ by taking

\[ \varphi _{ij} : x_ i|_{U_ i \times _ U U_ j} \longrightarrow x_ j|_{U_ i \times _ U U_ j} \]

the arrow such that $F(\varphi _{ij}) = \varphi _ j^{-1} \circ \varphi _ i$. This descent datum is effective by the axioms of a stack, and hence we obtain an object $x_1$ of $\mathcal{S}_1$ over $U$. We omit the verification that $F(x_1)$ is isomorphic to $x$ over $U$. $\square$

Remark 8.4.9. (Cutting down a “big” stack to get a stack.) Let $\mathcal{C}$ be a site. Suppose that $p : \mathcal{S} \to \mathcal{C}$ is functor from a “big” category to $\mathcal{C}$, i.e., suppose that the collection of objects of $\mathcal{S}$ forms a proper class. Finally, suppose that $p : \mathcal{S} \to \mathcal{C}$ satisfies conditions (1), (2), (3) of Definition 8.4.1. In general there is no way to replace $p : \mathcal{S} \to \mathcal{C}$ by a equivalent category such that we obtain a stack. The reason is that it can happen that a fibre categories $\mathcal{S}_ U$ may have a proper class of isomorphism classes of objects. On the other hand, suppose that

  1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a set $S_ U \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ such that every object of $\mathcal{S}_ U$ is isomorphic in $\mathcal{S}_ U$ to an element of $S_ U$.

In this case we can find a full subcategory $\mathcal{S}_{small}$ of $\mathcal{S}$ such that, setting $p_{small} = p|_{\mathcal{S}_{small}}$, we have

  1. the functor $p_{small} : \mathcal{S}_{small} \to \mathcal{C}$ defines a stack, and

  2. the inclusion $\mathcal{S}_{small} \to \mathcal{S}$ is fully faithful and essentially surjective.

(Hint: For every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ let $\alpha (U)$ denote the smallest ordinal such that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U) \cap V_{\alpha (U)}$ surjects onto the set of isomorphism classes of $\mathcal{S}_ U$, and set $\alpha = \sup _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} \alpha (U)$. Then take $\mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{small}) = \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}) \cap V_\alpha $. For notation used see Sets, Section 3.5.)


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