## 89.21 Groupoids in functors on an arbitrary category

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.

1. 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.

2. 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

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

Remark 89.21.2. A groupoid in functors on $\mathcal{C}$ amounts to the data of a functor $\mathcal{C} \to \textit{Groupoids}$, and a morphism of groupoids in functors on $\mathcal{C}$ amounts to a morphism of the corresponding functors $\mathcal{C} \to \textit{Groupoids}$ (where $\textit{Groupoids}$ is regarded as a 1-category). However, for our purposes it is more convenient to use the terminology of groupoids in functors. In fact, thinking of a groupoid in functors as the corresponding functor $\mathcal{C} \to \textit{Groupoids}$, or equivalently as the category cofibered in groupoids associated to that functor, can lead to confusion (Remark 89.23.2).

Remark 89.21.3. Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$. There are unique morphisms $e : U \to R$ and $i : R \to R$ such that for every object $T$ of $\mathcal{C}$, $e: U(T) \to R(T)$ sends $x \in U(T)$ to the identity morphism on $x$ and $i: R(T) \to R(T)$ sends $a \in U(T)$ to the inverse of $a$ in the groupoid category $(U(T), R(T), s, t, c)$. We will sometimes refer to $s$, $t$, $c$, $e$, and $i$ as “source”, “target”, “composition”, “identity”, and “inverse”.

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.

Remark 89.21.5. Hence a representable groupoid in functors on $\mathcal{C}$ is given by objects $U$ and $R$ of $\mathcal{C}$ and morphisms $s, t : U \to R$ and $c : R \to R \amalg _{s, U, t} R$ such that $(\underline{U}, \underline{R}, s, t, c)$ satisfies the condition of Definition 89.21.1. The reason for requiring the existence of the pushout $R \amalg _{s, U, t} R$ is so that the composition morphism $c$ is defined at the level of morphisms in $\mathcal{C}$. This requirement will always be satisfied below when we consider representable groupoids in functors on $\widehat{\mathcal{C}}_\Lambda$, since by Lemma 89.4.3 the category $\widehat{\mathcal{C}}_\Lambda$ admits pushouts.

Remark 89.21.6. We will say “let $(\underline{U}, \underline{R}, s, t, c)$ be a groupoid in functors on $\mathcal{C}$” to mean that we have a representable groupoid in functors. Thus this means that $U$ and $R$ are objects of $\mathcal{C}$, there are morphisms $s, t : U \to R$, the pushout $R \amalg _{s, U, t} R$ exists, there is a morphism $c : R \to R \amalg _{s, U, t} R$, and $(\underline{U}, \underline{R}, s, t, c)$ is a groupoid in functors on $\mathcal{C}$.

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 $(U|_{\mathcal{C}'}, R|_{\mathcal C'}, s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'})$.

Remark 89.21.8. In the situation of Definition 89.21.7, we often denote $s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'}$ simply by $s, t, c$.

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

1. 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)).

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

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

2. $x \in U(T)$ 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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