The Stacks project

Proof. Omitted. $\square$

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 $\mathcal{S}'$. $\square$

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 4.33.8 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 4.33.8 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

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 $i$. $\square$

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 4.32.3 is a fibred category according to Categories, Lemma 4.33.10. (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

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 4.32.3). 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. $\square$

Proof. Assume (1). For $U, x, y$ as in (2) the displayed map $F$ evaluates to the map $F : \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_{1, V}}(x|_ V, y|_ V) \to \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_{2, V}}(F(x|_ V), F(y|_ V))$ on an object $V$ of $\mathcal{C}$ lying over $U$. Now, since $F$ is fully faithful, the corresponding map $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_1}(x|_ V, y|_ V) \to \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_2}(F(x|_ V), F(y|_ V))$ 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 $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_{1, U}}(x, f^*y)$. 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 $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_{2, U}}(F(x), F(f^*y))$. Since $F$ induces a bijection $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_{1, U}}(x, f^*y) \to \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_{2, U}}(F(x), F(f^*y))$ we conclude (1) holds. $\square$

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

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 $U$. $\square$