The Stacks project

88.22 Groupoids in functors on the base category

In this section we discuss groupoids in functors on $\mathcal{C}_\Lambda $. Our eventual goal is to show that prorepresentable groupoids in functors on $\mathcal{C}_\Lambda $ serve as “presentations” for well-behaved deformation categories in the same way that smooth groupoids in algebraic spaces serve as presentations for algebraic stacks, cf. Algebraic Stacks, Section 92.16.

Definition 88.22.1. A groupoid in functors on $\mathcal{C}_\Lambda $ is prorepresentable if it is isomorphic to $(\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda }$ for some representable groupoid in functors $(\underline{R_0}, \underline{R_1}, s, t, c)$ on the category $\widehat{\mathcal{C}}_\Lambda $.

Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda $. Taking completions, we get a quintuple $(\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})$. By Remark 88.7.10 completion as a functor on $\text{CofSet}(\mathcal{C}_\Lambda )$ is a right adjoint, so it commutes with limits. In particular, there is a canonical isomorphism

\[ \widehat{R \times _{s, U, t} R} \longrightarrow \widehat{R} \times _{\widehat{s}, \widehat{U}, \widehat{t}} \widehat{R}, \]

so $\widehat{c}$ can be regarded as a functor $\widehat{R} \times _{\widehat{s}, \widehat{U}, \widehat{t}} \widehat{R} \to \widehat{R}$. Then $(\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})$ is a groupoid in functors on $\widehat{\mathcal{C}}_\Lambda $, with identity and inverse morphisms being the completions of those of $(U, R, s, t, c)$.

Definition 88.22.2. Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda $. The completion $(U, R, s, t, c)^{\wedge }$ of $(U, R, s, t, c)$ is the groupoid in functors $(\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})$ on $\widehat{\mathcal{C}}_\Lambda $ described above.

Remark 88.22.3. Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda $. Then there is a canonical isomorphism $(U, R, s, t, c)^{\wedge }|_{\mathcal{C}_\Lambda } \cong (U, R, s, t, c)$, see Remark 88.7.7. On the other hand, let $(U, R, s, t, c)$ be a groupoid in functors on $\widehat{\mathcal{C}}_\Lambda $ such that $U, R : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ both commute with limits, e.g. if $U, R$ are representable. Then there is a canonical isomorphism $((U, R, s, t, c)|_{\mathcal{C}_\Lambda })^{\wedge } \cong (U, R, s, t, c)$. This follows from Remark 88.7.11.

Lemma 88.22.4. 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 88.22.3. 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 88.7.11 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 88.4.3 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 88.22.3 hence $(U, R, s, t, c)$ is prorepresentable by definition. $\square$


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