Definition 7.14.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A morphism of sites $f : \mathcal{D} \to \mathcal{C}$ is given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$ such that the functor $u_ s$ is exact.
7.14 Morphisms of sites
Notice how the functor $u$ goes in the direction opposite the morphism $f$. If $f \leftrightarrow u$ is a morphism of sites then we use the notation $f^{-1} = u_ s$ and $f_* = u^ s$. The functor $f^{-1}$ is called the pullback functor and the functor $f_*$ is called the pushforward functor. As in topology we have the following adjointness property
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, f_*\mathcal{F}) \]
The motivation for this definition comes from the following example.
Example 7.14.2. Let $f : X \to Y$ be a continuous map of topological spaces. Recall that we have sites $X_{Zar}$ and $Y_{Zar}$, see Example 7.6.4. Consider the functor $u : Y_{Zar} \to X_{Zar}$, $V \mapsto f^{-1}(V)$. This functor is clearly continuous because inverse images of open coverings are open coverings. (Actually, this depends on how you chose sets of coverings for $X_{Zar}$ and $Y_{Zar}$. But in any case the functor is quasi-continuous, see Remark 7.13.6.) It is easy to verify that the functor $u^ s$ equals the usual pushforward functor $f_*$ from topology. Hence, since $u_ s$ is an adjoint and since the usual topological pullback functor $f^{-1}$ is an adjoint as well, we get a canonical isomorphism $f^{-1} \cong u_ s$. Since $f^{-1}$ is exact we deduce that $u_ s$ is exact. Hence $u$ defines a morphism of sites $f : X_{Zar} \to Y_{Zar}$, which we may denote $f$ as well since we've already seen the functors $u_ s, u^ s$ agree with their usual notions anyway.
Example 7.14.3. Let $\mathcal{C}$ be a category. Let be two sets of families of morphisms with fixed target which turn $\mathcal{C}$ into a site. Denote $\mathcal{C}_\tau $ the site corresponding to $\text{Cov}(\mathcal{C})$ and $\mathcal{C}_{\tau '}$ the site corresponding to $\text{Cov}'(\mathcal{C})$. We claim the identity functor on $\mathcal{C}$ defines a morphism of sites Namely, observe that $\text{id} : \mathcal{C}_{\tau '} \to \mathcal{C}_\tau $ is continuous as every $\tau '$-covering is a $\tau $-covering. Thus the functor $\epsilon _* = \text{id}^ s$ is the identity functor on underlying presheaves. Hence the left adjoint $\epsilon ^{-1}$ of $\epsilon _*$ sends a $\tau '$-sheaf $\mathcal{F}$ to the $\tau $-sheafification of $\mathcal{F}$ (by the universal property of sheafification). Finite limits of $\tau '$-sheaves agree with finite limits of presheaves (Lemma 7.10.1) and $\tau $-sheafification commutes with finite limits (Lemma 7.10.14). Thus $\epsilon ^{-1}$ is left exact. Since $\epsilon ^{-1}$ is a left adjoint it is also right exact (Categories, Lemma 4.24.6). Thus $\epsilon ^{-1}$ is exact and we have checked all the conditions of Definition 7.14.1.
\[ \text{Cov}(\mathcal{C}) \supset \text{Cov}'(\mathcal{C}) \]
\[ \epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau '} \]
Lemma 7.14.4. Let $\mathcal{C}_ i$, $i = 1, 2, 3$ be sites. Let $u : \mathcal{C}_2 \to \mathcal{C}_1$ and $v : \mathcal{C}_3 \to \mathcal{C}_2$ be continuous functors which induce morphisms of sites. Then the functor $u \circ v : \mathcal{C}_3 \to \mathcal{C}_1$ is continuous and defines a morphism of sites $\mathcal{C}_1 \to \mathcal{C}_3$.
Proof. It is immediate from the definitions that $u \circ v$ is a continuous functor. In addition, we clearly have $(u \circ v)^ p = v^ p \circ u^ p$, and hence $(u \circ v)^ s = v^ s \circ u^ s$. Hence functors $(u \circ v)_ s$ and $u_ s \circ v_ s$ are both left adjoints of $(u \circ v)^ s$. Therefore $(u \circ v)_ s \cong u_ s \circ v_ s$ and we conclude that $(u \circ v)_ s$ is exact as a composition of exact functors. $\square$
Definition 7.14.5. Let $\mathcal{C}_ i$, $i = 1, 2, 3$ be sites. Let $f : \mathcal{C}_1 \to \mathcal{C}_2$ and $g : \mathcal{C}_2 \to \mathcal{C}_3$ be morphisms of sites given by continuous functors $u : \mathcal{C}_2 \to \mathcal{C}_1$ and $v : \mathcal{C}_3 \to \mathcal{C}_2$. The composition $g \circ f$ is the morphism of sites corresponding to the functor $u \circ v$.
In this situation we have $(g \circ f)_* = g_* \circ f_*$ and $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ (see proof of Lemma 7.14.4).
Lemma 7.14.6. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be continuous. Assume all the categories $(\mathcal{I}_ V^ u)^{opp}$ of Section 7.5 are filtered. Then $u$ defines a morphism of sites $\mathcal{D} \to \mathcal{C}$, in other words $u_ s$ is exact.
Proof. Since $u_ s$ is the left adjoint of $u^ s$ we see that $u_ s$ is right exact, see Categories, Lemma 4.24.6. Hence it suffices to show that $u_ s$ is left exact. In other words we have to show that $u_ s$ commutes with finite limits. Because the categories $\mathcal{I}_ Y^{opp}$ are filtered we see that $u_ p$ commutes with finite limits, see Categories, Lemma 4.19.2 (this also uses the description of limits in $\textit{PSh}$, see Section 7.4). And since sheafification commutes with finite limits as well (Lemma 7.10.14) we conclude because $u_ s = \# \circ u_ p$. $\square$
Proposition 7.14.7. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be continuous. Assume furthermore the following:
the category $\mathcal{C}$ has a final object $X$ and $u(X)$ is a final object of $\mathcal{D}$ , and
the category $\mathcal{C}$ has fibre products and $u$ commutes with them.
Then $u$ defines a morphism of sites $\mathcal{D} \to \mathcal{C}$, in other words $u_ s$ is exact.
Remark 7.14.8. The conditions of Proposition 7.14.7 above are equivalent to saying that $u$ is left exact, i.e., commutes with finite limits. See Categories, Lemmas 4.18.4 and 4.23.2. It seems more natural to phrase it in terms of final objects and fibre products since this seems to have more geometric meaning in the examples.
Lemma 7.19.4 will provide another way to prove a continuous functor gives rise to a morphism of sites.
Remark 7.14.9. (Skip on first reading.) Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Analogously to Definition 7.14.1 we say that a quasi-morphism of sites $f : \mathcal{D} \to \mathcal{C}$ is given by a quasi-continuous functor $u : \mathcal{C} \to \mathcal{D}$ (see Remark 7.13.6) such that $u_ s$ is exact. The analogue of Proposition 7.14.7 in this setting is obtained by replacing the word “continuous” by the word “quasi-continuous”, and replacing the word “morphism” by “quasi-morphism”. The proof is literally the same.
In Definition 7.14.1 the condition that $u_ s$ be exact cannot be omitted. For example, the conclusion of the following lemma need not hold if one only assumes that $u$ is continuous.
Lemma 7.14.10. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by the functor $u : \mathcal{C} \to \mathcal{D}$. Given any object $V$ of $\mathcal{D}$ there exists a covering $\{ V_ j \to V\} $ such that for every $j$ there exists a morphism $V_ j \to u(U_ j)$ for some object $U_ j$ of $\mathcal{C}$.
Proof. Since $f^{-1} = u_ s$ is exact we have $f^{-1}* = *$ where $*$ denotes the final object of the category of sheaves (Example 7.10.2). Since $f^{-1}* = u_ s*$ is the sheafification of $u_ p*$ we see there exists a covering $\{ V_ j \to V\} $ such that $(u_ p*)(V_ j)$ is nonempty. Since $(u_ p*)(V_ j)$ is a colimit over the category $\mathcal{I}^ u_{V_ j}$ whose objects are morphisms $V_ j \to u(U)$ the lemma follows. $\square$
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