The Stacks project

Lemma 89.22.4. Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda $.

  1. $(U, R, s, t, c)$ is prorepresentable if and only if its completion is representable as a groupoid in functors on $\widehat{\mathcal{C}}_\Lambda $.

  2. $(U, R, s, t, c)$ is prorepresentable if and only if $U$ and $R$ are prorepresentable.

Proof. Part (1) follows from Remark 89.22.3. For (2), the “only if” direction is clear from the definition of a prorepresentable groupoid in functors. Conversely, assume $U$ and $R$ are prorepresentable, say $U \cong \underline{R_0}|_{\mathcal{C}_\Lambda }$ and $R \cong \underline{R_1}|_{\mathcal{C}_\Lambda }$ for objects $R_0$ and $R_1$ of $\widehat{\mathcal{C}}_\Lambda $. Since $\underline{R_0} \cong \widehat{\underline{R_0}|_{\mathcal{C}_\Lambda }}$ and $\underline{R_1} \cong \widehat{\underline{R_1}|_{\mathcal{C}_\Lambda }}$ by Remark 89.7.11 we see that the completion $(U, R, s, t, c)^\wedge $ is a groupoid in functors of the form $(\underline{R_0}, \underline{R_1}, \widehat{s}, \widehat{t}, \widehat{c})$. By Lemma 89.4.3 the pushout $\underline{R_1} \times _{\widehat{s}, \underline{R_1}, \widehat{t}} \underline{R_1}$ exists. Hence $(\underline{R_0}, \underline{R_1}, \widehat{s}, \widehat{t}, \widehat{c})$ is a representable groupoid in functors on $\widehat{\mathcal{C}}_\Lambda $. Finally, the restriction $(\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda }$ gives back $(U, R, s, t, c)$ by Remark 89.22.3 hence $(U, R, s, t, c)$ is prorepresentable by definition. $\square$

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