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40.3 Useful diagrams

We briefly restate the results of Groupoids, Lemmas 39.13.4 and 39.13.5 for easy reference in this chapter. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. In the commutative diagram
\begin{equation} \label{more-groupoids-equation-diagram} \vcenter { \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 } } \end{equation}

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

The diagram
\begin{equation} \label{more-groupoids-equation-pull} \vcenter { \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.

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