## 7.16 G-sets and morphisms

Let $\varphi : G \to H$ be a homomorphism of groups. Choose (suitable) sites $\mathcal{T}_ G$ and $\mathcal{T}_ H$ as in Example 7.6.5 and Section 7.9. Let $u : \mathcal{T}_ H \to \mathcal{T}_ G$ be the functor which assigns to a $H$-set $U$ the $G$-set $U_\varphi$ which has the same underlying set but $G$ action defined by $g \cdot \xi = \varphi (g)\xi$ for $g \in G$ and $\xi \in U$. It is clear that $u$ commutes with finite limits and is continuous1. Applying Proposition 7.14.7 and Lemma 7.15.2 we obtain a morphism of topoi

$f : \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ H)$

associated with $\varphi$. Using Proposition 7.9.1 we see that we get a pair of adjoint functors

$f_* : G\textit{-Sets} \longrightarrow H\textit{-Sets}, \quad f^{-1} : H\textit{-Sets} \longrightarrow G\textit{-Sets}.$

Let's work out what are these functors in this case.

We first work out a formula for $f_*$. Recall that given a $G$-set $S$ the corresponding sheaf $\mathcal{F}_ S$ on $\mathcal{T}_ G$ is given by the rule $\mathcal{F}_ S(U) = \mathop{\mathrm{Mor}}\nolimits _ G(U, S)$. And on the other hand, given a sheaf $\mathcal{G}$ on $\mathcal{T}_ H$ the corresponding $H$-set is given by the rule $\mathcal{G}({}_ HH)$. Hence we see that

$f_*S = \mathop{\mathrm{Mor}}\nolimits _{G\textit{-Sets}}(({}_ HH)_\varphi , S).$

If we work this out a little bit more then we get

$f_*S = \{ a : H \to S \mid a(gh) = ga(h) \}$

with left $H$-action given by $(h \cdot a)(h') = a(h'h)$ for any element $a \in f_*S$.

Next, we explicitly compute $f^{-1}$. Note that since the topology on $\mathcal{T}_ G$ and $\mathcal{T}_ H$ is subcanonical, all representable presheaves are sheaves. Moreover, given an object $V$ of $\mathcal{T}_ H$ we see that $f^{-1}h_ V$ is equal to $h_{u(V)}$ (see Lemma 7.13.5). Hence we see that $f^{-1}S = S_\varphi$ for representable sheaves. Since every sheaf on $\mathcal{T}_ H$ is a coproduct of representable sheaves we conclude that this is true in general. Hence we see that for any $H$-set $T$ we have

$f^{-1}T = T_\varphi .$

The adjunction between $f^{-1}$ and $f_*$ is evidenced by the formula

$\mathop{\mathrm{Mor}}\nolimits _{G\textit{-Sets}}(T_\varphi , S) = \mathop{\mathrm{Mor}}\nolimits _{H\textit{-Sets}}(T, f_*S)$

with $f_*S$ as above. This can be proved directly. Moreover, it is then clear that $(f^{-1}, f_*)$ form an adjoint pair and that $f^{-1}$ is exact. So alternatively to the above the morphism of topoi $f : G\textit{-Sets} \to H\textit{-Sets}$ can be defined directly in this manner.

[1] Set theoretical remark: First choose $\mathcal{T}_ H$. Then choose $\mathcal{T}_ G$ to contain $u(\mathcal{T}_ H)$ and such that every covering in $\mathcal{T}_ H$ corresponds to a covering in $\mathcal{T}_ G$. This is possible by Sets, Lemmas 3.10.1, 3.10.2 and 3.11.1.

Comment #7481 by Nicolás on

A small nitpick comment: in the first paragraph, the letter $u$ is used both to reference the functor $u\colon \mathcal{T}_H \to \mathcal{T}_G$ and some element $u \in U$.

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