7.13 Continuous functors
Definition 7.13.1. Let \mathcal{C} and \mathcal{D} be sites. A functor u : \mathcal{C} \to \mathcal{D} is called continuous if for every \{ V_ i \to V\} _{i\in I} \in \text{Cov}(\mathcal{C}) we have the following
\{ u(V_ i) \to u(V)\} _{i\in I} is in \text{Cov}(\mathcal{D}), and
for any morphism T \to V in \mathcal{C} the morphism u(T \times _ V V_ i) \to u(T) \times _{u(V)} u(V_ i) is an isomorphism.
Recall that given a functor u as above, and a presheaf of sets \mathcal{F} on \mathcal{D} we have defined u^ p\mathcal{F} to be simply the presheaf \mathcal{F} \circ u, in other words
u^ p\mathcal{F} (V) = \mathcal{F}(u(V))
for every object V of \mathcal{C}.
Lemma 7.13.2. Let \mathcal{C} and \mathcal{D} be sites. Let u : \mathcal{C} \to \mathcal{D} be a continuous functor. If \mathcal{F} is a sheaf on \mathcal{D} then u^ p\mathcal{F} is a sheaf as well.
Proof.
Let \{ V_ i \to V\} be a covering. By assumption \{ u(V_ i) \to u(V)\} is a covering in \mathcal{D} and u(V_ i \times _ V V_ j) = u(V_ i)\times _{u(V)}u(V_ j). Hence the sheaf condition for u^ p\mathcal{F} and the covering \{ V_ i \to V\} is precisely the same as the sheaf condition for \mathcal{F} and the covering \{ u(V_ i) \to u(V)\} .
\square
In order to avoid confusion we sometimes denote
u^ s : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C})
the functor u^ p restricted to the subcategory of sheaves of sets. Recall that u^ p has a left adjoint u_ p : \textit{PSh}(\mathcal{C}) \to \textit{PSh}(\mathcal{D}), see Section 7.5.
Lemma 7.13.3. In the situation of Lemma 7.13.2. The functor u_ s : \mathcal{G} \mapsto (u_ p \mathcal{G})^\# is a left adjoint to u^ s.
Proof.
Follows directly from Lemma 7.5.4 and Proposition 7.10.12.
\square
Here is a technical lemma.
Lemma 7.13.4. In the situation of Lemma 7.13.2. For any presheaf \mathcal{G} on \mathcal{C} we have (u_ p\mathcal{G})^\# = (u_ p(\mathcal{G}^\# ))^\# .
Proof.
For any sheaf \mathcal{F} on \mathcal{D} we have
\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(u_ s(\mathcal{G}^\# ), \mathcal{F}) & = & \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}^\# , u^ s\mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{G}^\# , u^ p\mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{G}, u^ p\mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(u_ p\mathcal{G}, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}((u_ p\mathcal{G})^\# , \mathcal{F}) \end{eqnarray*}
and the result follows from the Yoneda lemma.
\square
Lemma 7.13.5. Let u : \mathcal{C} \to \mathcal{D} be a continuous functor between sites. For any object U of \mathcal{C} we have u_ sh_ U^\# = h_{u(U)}^\# .
Proof.
Follows from Lemmas 7.5.6 and 7.13.4.
\square
Comments (2)
Comment #3520 by Laurent Moret-Bailly on
Comment #3660 by Johan on