Definition 90.6.1. Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. We say $F$ is prorepresentable if there exists an isomorphism $F \cong \underline{R}|_{\mathcal{C}_\Lambda }$ of functors for some $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$.
90.6 Prorepresentable functors and predeformation categories
Our basic goal is to understand categories cofibered in groupoids over $\mathcal{C}_\Lambda $ and $\widehat{\mathcal{C}}_\Lambda $. Since $\mathcal{C}_\Lambda $ is a full subcategory of $\widehat{\mathcal{C}}_\Lambda $ we can restrict categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda $ to $\mathcal{C}_\Lambda $, see Remarks 90.5.2 (15). In particular we can do this with functors, in particular with representable functors. The functors on $\mathcal{C}_\Lambda $ one obtains in this way are called prorepresentable functors.
Note that if $F : \mathcal{C}_\Lambda \to \textit{Sets}$ is prorepresentable by $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$, then
\[ F(k) = \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(R, k) = \{ *\} \]
is a singleton. The categories cofibered in groupoids over $\mathcal{C}_\Lambda $ that are arise in deformation theory will often satisfy an analogous condition.
Definition 90.6.2. A predeformation category $\mathcal{F}$ is a category cofibered in groupoids over $\mathcal{C}_\Lambda $ such that $\mathcal{F}(k)$ is equivalent to a category with a single object and a single morphism, i.e., $\mathcal{F}(k)$ contains at least one object and there is a unique morphism between any two objects. A morphism of predeformation categories is a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda $.
A feature of a predeformation category is the following. Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Then every object of $\mathcal{F}$ comes equipped with a unique morphism to $x_0$. Namely, if $x$ is an object of $\mathcal{F}$ over $A$, then we can choose a pushforward $x \to q_*x$ where $q : A \to k$ is the quotient map. There is a unique isomorphism $q_*x \to x_0$ and the composition $x \to q_*x \to x_0$ is the desired morphism.
Remark 90.6.3. We say that a functor $F: \mathcal{C}_\Lambda \to \textit{Sets}$ is a predeformation functor if the associated cofibered set is a predeformation category, i.e. if $F(k)$ is a one element set. Thus if $\mathcal{F}$ is a predeformation category, then $\overline{\mathcal{F}}$ is a predeformation functor.
Remark 90.6.4. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ be a category cofibered in groupoids, and let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. We denote by $\mathcal{F}_ x$ the category of objects over $x$. An object of $\mathcal{F}_ x$ is an arrow $y \to x$. A morphism $(y \to x) \to (z \to x)$ in $\mathcal{F}_ x$ is a commutative diagram There is a forgetful functor $\mathcal{F}_ x \to \mathcal{F}$. We define the functor $p_ x : \mathcal{F}_ x \to \mathcal{C}_\Lambda $ as the composition $\mathcal{F}_ x \to \mathcal{F} \xrightarrow {p} \mathcal{C}_\Lambda $. Then $p_ x : \mathcal{F}_ x \to \mathcal{C}_\Lambda $ is a predeformation category (proof omitted). In this way we can pass from an arbitrary category cofibered in groupoids over $\mathcal{C}_\Lambda $ to a predeformation category at any $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$.
\[ \xymatrix{ y \ar[rr] \ar[dr] & & z \ar[dl] \\ & x & } \]
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