Lemma 78.23.1 (044T)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.23: The 2-coequalizer property of a quotient stack
Lemma 78.23.1 (cite)

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Lemma 78.23.1. Assumptions and notation as in Lemmas 78.20.2 and 78.20.3. The vertical composition of

\[ \xymatrix@C=15pc{ \mathcal{S}_{R \times _{s, U, t} R} \ruppertwocell ^{\pi \circ s \circ \text{pr}_1 = \pi \circ s \circ c}{\ \ \ \ \ \ \alpha \star \text{id}_{\text{pr}_1}} \ar[r]_(.3){\pi \circ t \circ \text{pr}_1 = \pi \circ s \circ \text{pr}_0} \rlowertwocell _{\pi \circ t \circ \text{pr}_0 = \pi \circ t \circ c}{\ \ \ \ \ \ \alpha \star \text{id}_{\text{pr}_0}} & [U/R] } \]

is the $2$-morphism $\alpha \star \text{id}_ c$. In a formula $\alpha \star \text{id}_ c = (\alpha \star \text{id}_{\text{pr}_0}) \circ (\alpha \star \text{id}_{\text{pr}_1}) $.

Proof. We make two remarks:

The formula $\alpha \star \text{id}_ c = (\alpha \star \text{id}_{\text{pr}_0}) \circ (\alpha \star \text{id}_{\text{pr}_1})$ only makes sense if you realize the equalities $\pi \circ s \circ \text{pr}_1 = \pi \circ s \circ c$, $\pi \circ t \circ \text{pr}_1 = \pi \circ s \circ \text{pr}_0$, and $\pi \circ t \circ \text{pr}_0 = \pi \circ t \circ c$. Namely, the second one implies the vertical composition $\circ $ makes sense, and the other two guarantee the two sides of the formula are $2$-morphisms with the same source and target.
The reason the lemma holds is that composition in the category fibred in groupoids $[U/_{\! p}R]$ associated to the presheaf in groupoids (78.20.0.1) comes from the composition law $c : R \times _{s, U, t} R \to R$.

We omit the proof of the lemma. $\square$

The Stacks project

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