## 78.11 Groupoids in algebraic spaces

Please refer to Groupoids, Section 39.13 for notation.

Definition 78.11.1. Let $B \to S$ as in Section 78.3.

A *groupoid in algebraic spaces over $B$* is a quintuple $(U, R, s, t, c)$ where $U$ and $R$ are algebraic spaces over $B$, and $s, t : R \to U$ and $c : R \times _{s, U, t} R \to R$ are morphisms of algebraic spaces over $B$ with the following property: For any scheme $T$ over $B$ the quintuple

\[ (U(T), R(T), s, t, c) \]

is a groupoid category.

A *morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in algebraic spaces over $B$* is given by morphisms of algebraic spaces $f : U \to U'$ and $f : R \to R'$ over $B$ with the following property: For any scheme $T$ over $B$ the maps $f$ define a functor from the groupoid category $(U(T), R(T), s, t, c)$ to the groupoid category $(U'(T), R'(T), s', t', c')$.

Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Note that there are unique morphisms of algebraic spaces $e : U \to R$ and $i : R \to R$ over $B$ such that for every scheme $T$ over $B$ the induced map $e : U(T) \to R(T)$ is the identity, and $i : R(T) \to R(T)$ is the inverse of the groupoid category. The septuple $(U, R, s, t, c, e, i)$ satisfies commutative diagrams corresponding to each of the axioms (1), (2)(a), (2)(b), (3)(a) and (3)(b) of Groupoids, Section 39.13. Conversely given a septuple with this property the quintuple $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $B$. Note that $i$ is an isomorphism, and $e$ is a section of both $s$ and $t$. Moreover, given a groupoid in algebraic spaces over $B$ we denote

\[ j = (t, s) : R \longrightarrow U \times _ B U \]

which is compatible with our conventions in Section 78.4 above. We sometimes say “let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$” to stress the existence of identity and inverse.

Lemma 78.11.2. Let $B \to S$ as in Section 78.3. Given a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ the morphism $j : R \to U \times _ B U$ is a pre-equivalence relation.

**Proof.**
Omitted. This is a nice exercise in the definitions.
$\square$

Lemma 78.11.3. Let $B \to S$ as in Section 78.3. Given an equivalence relation $j : R \to U \times _ B U$ over $B$ there is a unique way to extend it to a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$.

**Proof.**
Omitted. This is a nice exercise in the definitions.
$\square$

Lemma 78.11.4. Let $B \to S$ as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. In the commutative diagram

\[ \xymatrix{ & U & \\ R \ar[d]_ s \ar[ru]^ t & R \times _{s, U, t} R \ar[l]^-{\text{pr}_0} \ar[d]^{\text{pr}_1} \ar[r]_-c & R \ar[d]^ s \ar[lu]_ t \\ U & R \ar[l]_ t \ar[r]^ s & U } \]

the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian.

**Proof.**
Omitted. Exercise in the definitions and the functorial point of view in algebraic geometry.
$\square$

Lemma 78.11.5. Let $B \to S$ be as in Section 78.3. Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$. The diagram

78.11.5.1
\begin{equation} \label{spaces-groupoids-equation-pull} \xymatrix{ R \times _{t, U, t} R \ar@<1ex>[r]^-{\text{pr}_1} \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{\text{pr}_0 \times c \circ (i, 1)} & R \ar[r]^ t \ar[d]^{\text{id}_ R} & U \ar[d]^{\text{id}_ U} \\ R \times _{s, U, t} R \ar@<1ex>[r]^-c \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{\text{pr}_1} & R \ar[r]^ t \ar[d]^ s & U \\ R \ar@<1ex>[r]^ s \ar@<-1ex>[r]_ t & U } \end{equation}

is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian.

**Proof.**
The commutativity of the diagram follows from the axioms of a groupoid. Note that, in terms of groupoids, the top left vertical arrow assigns to a pair of morphisms $(\alpha , \beta )$ with the same target, the pair of morphisms $(\alpha , \alpha ^{-1} \circ \beta )$. In any groupoid this defines a bijection between $\text{Arrows} \times _{t, \text{Ob}, t} \text{Arrows}$ and $\text{Arrows} \times _{s, \text{Ob}, t} \text{Arrows}$. Hence the second assertion of the lemma. The last assertion follows from Lemma 78.11.4.
$\square$

Lemma 78.11.6. Let $B \to S$ be as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $B' \to B$ be a morphism of algebraic spaces. Then the base changes $U' = B' \times _ B U$, $R' = B' \times _ B R$ endowed with the base changes $s'$, $t'$, $c'$ of the morphisms $s, t, c$ form a groupoid in algebraic spaces $(U', R', s', t', c')$ over $B'$ and the projections determine a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoids in algebraic spaces over $B$.

**Proof.**
Omitted. Hint: $R' \times _{s', U', t'} R' = B' \times _ B (R \times _{s, U, t} R)$.
$\square$

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