The Stacks project

90.21 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 90.21.1. 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 90.21.2. 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 90.23.2).

Remark 90.21.3. 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 90.21.4. 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 90.21.5. 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 90.21.1. 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 90.4.3 the category $\widehat{\mathcal{C}}_\Lambda $ admits pushouts.

Remark 90.21.6. 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 90.21.7. 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 90.21.8. In the situation of Definition 90.21.7, we often denote $s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'}$ simply by $s, t, c$.

Definition 90.21.9. 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 90.5.2 (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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