The Stacks project

Proof. Our definition of composition of morphisms of sites implies that $u_ b \circ u_ a = u_ c$ whenever $c = a \circ b$ in $\mathcal{I}$. The formula $\mathcal{C} = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i$ means that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ i)$ and $\text{Arrows}(\mathcal{C}) = \mathop{\mathrm{colim}}\nolimits \text{Arrows}(\mathcal{C}_ i)$. Then source, target, and composition are inherited from the source, target, and composition on $\text{Arrows}(\mathcal{C}_ i)$. In this way we obtain a category. Denote $u_ i : \mathcal{C}_ i \to \mathcal{C}$ the obvious functor. Remark that given any finite diagram in $\mathcal{C}$ there exists an $i$ such that this diagram is the image of a diagram in $\mathcal{C}_ i$.

Let $\{ U^ t \to U\} $ be a covering of $\mathcal{C}$. We first prove that if $V \to U$ is a morphism of $\mathcal{C}$, then $U^ t \times _ U V$ exists. By our remark above and our definition of coverings, we can find an $i$, a covering $\{ U_ i^ t \to U_ i\} $ of $\mathcal{C}_ i$ and a morphism $V_ i \to U_ i$ whose image by $u_ i$ is the given data. We claim that $U^ t \times _ U V$ is the image of $U^ t_ i \times _{U_ i} V_ i$ by $u_ i$. Namely, for every $a : j \to i$ in $\mathcal{I}$ the functor $u_ a$ is continuous, hence $u_ a(U^ t_ i \times _{U_ i} V_ i) = u_ a(U^ t_ i) \times _{u_ a(U_ i)} u_ a(V_ i)$. In particular we can replace $i$ by $j$, if we so desire. Thus, if $W$ is another object of $\mathcal{C}$, then we may assume $W = u_ i(W_ i)$ and we see that

as filtered colimits commute with finite limits (Categories, Lemma 4.19.2). It also follows that $\{ U^ t \times _ U V \to V\} $ is a covering in $\mathcal{C}$. In this way we see that axiom (3) of Definition 7.6.2 holds.

To verify axiom (2) of Definition 7.6.2 let $\{ U^ t \to U\} _{t \in T}$ be a covering of $\mathcal{C}$ and for each $t$ let $\{ U^{ts} \to U^ t\} $ be a covering of $\mathcal{C}$. Then we can find an $i$ and a covering $\{ U^ t_ i \to U_ i\} _{t \in T}$ of $\mathcal{C}_ i$ whose image by $u_ i$ is $\{ U^ t \to U\} $. Since $T$ is finite we may choose an $a : j \to i$ in $\mathcal{I}$ and coverings $\{ U^{ts}_ j \to u_ a(U^ t_ i)\} $ of $\mathcal{C}_ j$ whose image by $u_ j$ gives $\{ U^{ts} \to U^ t\} $. Then we conclude that $\{ U^{ts} \to U\} $ is a covering of $\mathcal{C}$ by an application of axiom (2) to the site $\mathcal{C}_ j$.

We omit the proof of axiom (1) of Definition 7.6.2. $\square$

Proof. It is immediate from the arguments in the proof of Lemma 7.18.2 that the functors $u_ i$ are continuous. To finish the proof we have to show that $f_ i^{-1} := u_{i, s}$ is an exact functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ i) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. In fact it suffices to show that $f_ i^{-1}$ is left exact, because it is right exact as a left adjoint (Categories, Lemma 4.24.6). We first prove (7.18.3.1) and then we deduce exactness.

For an arbitrary object $V$ of $\mathcal{C}$ we can pick a $a : j \to i$ and an object $V_ j \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ with $V = u_ j(V_ j)$. Then we can set

The value $\mathcal{G}(V)$ of the colimit is independent of the choice of $b : j \to i$ and of the object $V_ j$ with $u_ j(V_ j) = V$; we omit the verification. Moreover, if $\alpha : V \to V'$ is a morphism of $\mathcal{C}$, then we can choose $b : j \to i$ and a morphism $\alpha _ j : V_ j \to V'_ j$ with $u_ j(\alpha _ j) = \alpha $. This induces a map $\mathcal{G}(V') \to \mathcal{G}(V)$ by using the restrictions along the morphisms $u_ b(\alpha _ j) : u_ b(V_ j) \to u_ b(V'_ j)$. A check shows that $\mathcal{G}$ is a presheaf (omitted). In fact, $\mathcal{G}$ satisfies the sheaf condition. Namely, any covering $\mathcal{U} = \{ U^ t \to U\} $ in $\mathcal{C}$ comes from a finite level. Say $\mathcal{U}_ j = \{ U^ t_ j \to U_ j\} $ is mapped to $\mathcal{U}$ by $u_ j$ for some $a : j \to i$ in $\mathcal{I}$. Then we have

as desired. The first equality holds because filtered colimits commute with finite limits (Categories, Lemma 4.19.2). By construction $\mathcal{G}(U)$ is given by the right hand side of (7.18.3.1). Hence (7.18.3.1) is true if we can show that $\mathcal{G}$ is equal to $f_ i^{-1}\mathcal{F}$.

In this paragraph we check that $\mathcal{G}$ is canonically isomorphic to $f_ i^{-1}\mathcal{F}$. We strongly encourage the reader to skip this paragraph. To check this we have to show there is a bijection $\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, \mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ i)}(\mathcal{F}, f_{i, *}\mathcal{H})$ functorial in the sheaf $\mathcal{H}$ on $\mathcal{C}$ where $f_{i, *} = u_ i^ p$. A map $\mathcal{G} \to \mathcal{H}$ is the same thing as a compatible system of maps

for all $a : j \to i$, $b : k \to j$ and $V_ j \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ j)$. The compatibilities force the maps $\varphi _{a, b, V_ j}$ to be equal to $\varphi _{a \circ b, \text{id}, u_ b(V_ j)}$. Given $a : j \to i$, the family of maps $\varphi _{a, \text{id}, V_ j}$ corresponds to a map of sheaves $\varphi _ a : f_ a^{-1}\mathcal{F} \to f_{j, *}\mathcal{H}$. The compatibilities between the $\varphi _{a, \text{id}, u_ a(V_ i)}$ and the $\varphi _{\text{id}, \text{id}, V_ i}$ implies that $\varphi _ a$ is the adjoint of the map $\varphi _{id}$ via

Thus finally we see that the whole system of maps $\varphi _{a, b, V_ j}$ is determined by the map $\varphi _{id} : \mathcal{F} \to f_{i, *}\mathcal{H}$. Conversely, given such a map $\psi : \mathcal{F} \to f_{i, *}\mathcal{H}$ we can read the argument just given backwards to construct the family of maps $\varphi _{a, b, V_ j}$. This finishes the proof that $\mathcal{G} = f_ i^{-1}\mathcal{F}$.

Assume (7.18.3.1) holds. Then the functor $\mathcal{F} \mapsto f_ i^{-1}\mathcal{F}(U)$ commutes with finite limits because finite limits of sheaves are computed in the category of presheaves (Lemma 7.10.1), the functors $f_ a^{-1}$ commutes with finite limits, and filtered colimits commute with finite limits. To see that $\mathcal{F} \mapsto f_ i^{-1}\mathcal{F}(V)$ commutes with finite limits for a general object $V$ of $\mathcal{C}$, we can use the same argument using the formula for $f_ i^{-1}\mathcal{F}(V) = \mathcal{G}(V)$ given above. Thus $f_ i^{-1}$ is left exact and the proof of the lemma is complete. $\square$

Proof. A formal argument shows that

By (7.18.3.1) we see that the inner colimit is equal to $f_ j^{-1}\mathcal{F}_ j(u_ i(X_ i))$ hence we conclude by Lemma 7.17.7. $\square$

Proof. For the morphism

we choose the adjoint to the identity map

Hence $\varphi _ a$ is the counit for the adjunction given by $(f_ a^{-1}, f_{a, *})$. We must prove that for all $a : j \to i$ and $b : k \to i$ with composition $c = a \circ b$ we have $\varphi _ c = \varphi _ b \circ f_ b^{-1}\varphi _ a$. This follows from Categories, Lemma 4.24.9. Lastly, we must prove that the map given by adjunction

is an isomorphism. For an object $U$ of $\mathcal{C}$ we need to show the map

is bijective. Choose an $i$ and an object $U_ i$ of $\mathcal{C}_ i$ with $u_ i(U_ i) = U$. Then the left hand side is equal to

by Lemma 7.18.4. Since $u_ j(u_ a(U_ i)) = U$ we have $f_{j, *}\mathcal{F}(u_ a(U_ i)) = \mathcal{F}(U)$ for all $a : j \to i$ by definition. Hence the value of the colimit is $\mathcal{F}(U)$ and the proof is complete. $\square$