The Stacks project

77.17 Restricting groupoids

Please refer to Groupoids, Section 39.18 for notation.

Lemma 77.17.1. Let $B \to S$ as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \to U$ be a morphism of algebraic spaces. 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 in algebraic spaces over $B$ and such that $U' \to U$, $R' \to R$ defines a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoids in algebraic spaces over $B$. Moreover, for any scheme $T$ over $B$ the functor of groupoids

\[ (U'(T), R'(T), s', t', c') \to (U(T), R(T), s, t, c) \]

is the restriction (see Groupoids, Section 39.18) of $(U(T), R(T), s, t, c)$ via the map $U'(T) \to U(T)$.

Proof. Omitted. $\square$

Definition 77.17.2. Let $B \to S$ as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \to U$ be a morphism of algebraic spaces over $B$. The morphism of groupoids in algebraic spaces $(U', R', s', t', c') \to (U, R, s, t, c)$ constructed in Lemma 77.17.1 is called the restriction of $(U, R, s, t, c)$ to $U'$. We sometime use the notation $R' = R|_{U'}$ in this case.

Proof. What we are saying here is that $R'$ of Lemma 77.17.1 is also equal to

\[ R' = (U' \times _ B U')\times _{U \times _ B U} R \longrightarrow U' \times _ B U' \]

In fact this might have been a clearer way to state that lemma. $\square$


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