The Stacks project

Example 7.41.5. We construct a morphism $f : \mathcal{D} \to \mathcal{C}$ satisfying the assumptions of Lemma 7.41.4. Namely, let $\varphi : G \to H$ be a morphism of finite groups. Consider the sites $\mathcal{D} = \mathcal{T}_ G$ and $\mathcal{C} = \mathcal{T}_ H$ of countable $G$-sets and $H$-sets and coverings countable families of jointly surjective maps (Example 7.6.5). Let $u : \mathcal{T}_ H \to \mathcal{T}_ G$ be the functor described in Section 7.16 and $f : \mathcal{T}_ G \to \mathcal{T}_ H$ the corresponding morphism of sites. If $\varphi $ is injective, then every countable $G$-set is, as a $G$-set, the quotient of a countable $H$-set (this fails if $\varphi $ isn't injective). Thus $f$ satisfies the hypothesis of Lemma 7.41.4. If the sheaf $\mathcal{F}$ on $\mathcal{T}_ G$ corresponds to the $G$-set $S$, then the canonical map

\[ f^{-1}f_*\mathcal{F} \longrightarrow \mathcal{F} \]

corresponds to the map

\[ \text{Map}_ G(H, S) \longrightarrow S,\quad a \longmapsto a(1_ H) \]

If $\varphi $ is injective but not surjective, then this map is surjective (as it should according to Lemma 7.41.4) but not injective in general (for example take $G = \{ 1\} $, $H = \{ 1, \sigma \} $, and $S = \{ 1, 2\} $). Moreover, the functor $f_*$ does not commute with coequalizers or pushouts (for $G = \{ 1\} $ and $H = \{ 1, \sigma \} $).

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