The category of sheaves on a site is cartesian closed
Lemma 7.26.2. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be sheaves on $\mathcal{C}$. There is a canonical bijection which is functorial in all three entries.
Proof.
The lemma says that the functors $-\times \mathcal{G}$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},-)$ are adjoint to each other. To show this, we use the notion of unit and counit, see Categories, Section 4.24. The unit
sends $s \in \mathcal{F}(U)$ to the map $\mathcal{G}|_{\mathcal{C}/U} \to \mathcal{F}|_{\mathcal{C}/U}\times \mathcal{G}|_{\mathcal{C}/U}$ which over $V/U$ is given by
The counit
is the evaluation map. It is given by the rule
Then for each $\varphi : \mathcal{F} \times \mathcal{G} \to \mathcal{H}$, the corresponding morphism $\mathcal{F} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{H})$ is given by mapping each section $s \in \mathcal{F}(U)$ to the morphism of sheaves on $\mathcal{C}/U$ which on sections over $V/U$ is given by
Conversely, for each $\psi : \mathcal{F} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \mathcal{H})$, the corresponding morphism $\mathcal{F} \times \mathcal{G} \to \mathcal{H}$ is given by
on sections over an object $U$. We omit the details of the proof showing that these constructions are mutually inverse.
$\square$
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}\times \mathcal{G},\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F},\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{H})) \]
\[ \eta _\mathcal {F} : \mathcal{F} \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{F}\times \mathcal{G}) \]
\[ \mathcal{G}(V) \longrightarrow \mathcal{F}(V)\times \mathcal{G}(V), \quad t \longmapsto (s|_{V},t). \]
\[ \epsilon _{\mathcal{H}} : \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \mathcal{H}) \times \mathcal{G} \longrightarrow \mathcal{H} \]
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}( \mathcal{G}|_{\mathcal{C}/U}, \mathcal{H}|_{\mathcal{C}/U}) \times \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\quad (\varphi , s) \longmapsto \varphi (s). \]
\[ \mathcal{G}(V) \longrightarrow \mathcal{H}(V),\quad t \longmapsto \varphi (s|_ V, t). \]
\[ \mathcal{F}(U) \times \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\quad (s, t) \longmapsto \psi (s)(t) \]
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (1)
Comment #2785 by Stanisław Szawiel on