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

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 $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)