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

$\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}\times \mathcal{G},\mathcal{H}) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F},\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{H}))$

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

$\eta _\mathcal {F} : \mathcal{F} \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{F}\times \mathcal{G})$

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

$\mathcal{G}(V) \longrightarrow \mathcal{F}(V)\times \mathcal{G}(V), \quad t \longmapsto (s|_{V},t).$

The counit

$\epsilon _{\mathcal{H}} : \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \mathcal{H}) \times \mathcal{G} \longrightarrow \mathcal{H}$

is the evaluation map. It is given by the rule

$\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}( \mathcal{G}|_{\mathcal{C}/U}, \mathcal{H}|_{\mathcal{C}/U}) \times \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\quad (\varphi , s) \longmapsto \varphi (s).$

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

$\mathcal{G}(V) \longrightarrow \mathcal{H}(V),\quad t \longmapsto \varphi (s|_ V, t).$

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

$\mathcal{F}(U) \times \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\quad (s, t) \longmapsto \psi (s)(t)$

on sections over an object $U$. We omit the details of the proof showing that these constructions are mutually inverse. $\square$

Comment #2785 by StanisÅ‚aw Szawiel on

Suggested slogan: The category of sheaves on a site is cartesian closed.

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