39.18 Restricting groupoids
Consider a (usual) groupoid $\mathcal{C} = (\text{Ob}, \text{Arrows}, s, t, c)$. Suppose we have a map of sets $g : \text{Ob}' \to \text{Ob}$. Then we can construct a groupoid $\mathcal{C}' = (\text{Ob}', \text{Arrows}', s', t', c')$ by thinking of a morphism between elements $x', y'$ of $\text{Ob}'$ as a morphism in $\mathcal{C}$ between $g(x'), g(y')$. In other words we set
\[ \text{Arrows}' = \text{Ob}' \times _{g, \text{Ob}, t} \text{Arrows} \times _{s, \text{Ob}, g} \text{Ob}'. \]
with obvious choices for $s'$, $t'$, and $c'$. There is a canonical functor $\mathcal{C}' \to \mathcal{C}$ which is fully faithful, but not necessarily essentially surjective. This groupoid $\mathcal{C}'$ endowed with the functor $\mathcal{C}' \to \mathcal{C}$ is called the restriction of the groupoid $\mathcal{C}$ to $\text{Ob}'$.
Lemma 39.18.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Consider the following diagram
\[ \xymatrix{ R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'}& R \times _{s, U} U' \ar[r] \ar[d] & U' \ar[d]^ g \\ U' \times _{U, t} R \ar[d] \ar[r] & R \ar[r]^ s \ar[d]_ t & U \\ U' \ar[r]^ g & U } \]
where all the squares are fibre product squares. Then there is a canonical composition law $c' : R' \times _{s', U', t'} R' \to R'$ such that $(U', R', s', t', c')$ is a groupoid scheme over $S$ and such that $U' \to U$, $R' \to R$ defines a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes over $S$. Moreover, for any scheme $T$ over $S$ the functor of groupoids
\[ (U'(T), R'(T), s', t', c') \to (U(T), R(T), s, t, c) \]
is the restriction (see above) of $(U(T), R(T), s, t, c)$ via the map $U'(T) \to U(T)$.
Proof.
Omitted.
$\square$
Definition 39.18.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. The morphism of groupoids $(U', R', s', t', c') \to (U, R, s, t, c)$ constructed in Lemma 39.18.1 is called the restriction of $(U, R, s, t, c)$ to $U'$. We sometime use the notation $R' = R|_{U'}$ in this case.
Lemma 39.18.3. The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions 39.18.2 and 39.3.3 agree via the constructions of Lemmas 39.13.2 and 39.13.3.
Proof.
What we are saying here is that $R'$ of Lemma 39.18.1 is also equal to
\[ R' = (U' \times _ S U')\times _{U \times _ S U} R \longrightarrow U' \times _ S U' \]
In fact this might have been a clearer way to state that lemma.
$\square$
Lemma 39.18.4. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$. Let $G$ be the stabilizer of $(U, R, s, t, c)$ and let $G'$ be the stabilizer of $(U', R', s', t', c')$. Then $G'$ is the base change of $G$ by $g$, i.e., there is a canonical identification $G' = U' \times _{g, U} G$.
Proof.
Omitted.
$\square$
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