We begin with generalities on groupoids in functors on an arbitrary category. In the next section we will pass to the category $\mathcal{C}_\Lambda $. For clarity we shall sometimes refer to an ordinary groupoid, i.e., a category whose morphisms are all isomorphisms, as a groupoid category.

Definition 89.21.1. Let $\mathcal{C}$ be a category. The *category of groupoids in functors on $\mathcal{C}$* is the category with the following objects and morphisms.

Objects: A *groupoid in functors on $\mathcal{C}$* is a quintuple $(U, R, s, t, c)$ where $U, R : \mathcal{C} \to \textit{Sets}$ are functors and $s, t : R \to U$ and $c : R \times _{s, U, t} R \to R$ are morphisms with the following property: For any object $T$ of $\mathcal{C}$, the quintuple

\[ (U(T), R(T), s, t, c) \]

is a groupoid category.

Morphisms: A *morphism $(U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in functors on $\mathcal{C}$* consists of morphisms $U \to U'$ and $R \to R'$ with the following property: For any object $T$ of $\mathcal{C}$, the induced maps $U(T) \to U'(T)$ and $R(T) \to R'(T)$ define a functor between groupoid categories

\[ (U(T), R(T), s, t, c) \to (U'(T), R'(T), s', t', c'). \]

Definition 89.21.4. Let $\mathcal{C}$ be a category. A groupoid in functors on $\mathcal{C}$ is *representable* if it is isomorphic to one of the form $(\underline{U}, \underline{R}, s, t, c)$ where $U$ and $R$ are objects of $\mathcal{C}$ and the pushout $R \amalg _{s, U, t} R$ exists.

We introduce notation for restriction of groupoids in functors. This will be relevant below in situations where we restrict from $\widehat{\mathcal C}_\Lambda $ to $\mathcal{C}_\Lambda $.

Definition 89.21.7. Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$. Let $\mathcal{C}'$ be a subcategory of $\mathcal{C}$. The *restriction $(U, R, s, t, c)|_{\mathcal{C}'}$ of $(U, R, s, t, c)$ to $\mathcal{C}'$* is the groupoid in functors on $\mathcal{C}'$ given by $(U|_{\mathcal{C}'}, R|_{\mathcal C'}, s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'})$.

Definition 89.21.9. Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$.

The assignment $T \mapsto (U(T), R(T), s, t, c)$ determines a functor $\mathcal{C} \to \textit{Groupoids}$. The *quotient category cofibered in groupoids $[U/R] \to \mathcal{C}$* is the category cofibered in groupoids over $\mathcal{C}$ associated to this functor (as in Remarks 89.5.2 (9)).

The *quotient morphism $U \to [U/R]$* is the morphism of categories cofibered in groupoids over $\mathcal{C}$ defined by the rules

$x \in U(T)$ maps to the object $(T, x) \in \mathop{\mathrm{Ob}}\nolimits ([U/R](T))$, and

$x \in U(T)$ and $f : T \to T'$ give rise to the morphism $(f, \text{id}_{U(f)(x)}): (T, x) \to (T, U(f)(x))$ lying over $f : T \to T'$.

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