The Stacks project

88.25 Presentations of categories cofibered in groupoids

A presentation is defined as follows.

Definition 88.25.1. Let $\mathcal{F}$ be a category cofibered in groupoids over a category $\mathcal{C}$. Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}$. A presentation of $\mathcal{F}$ by $(U, R, s, t, c)$ is an equivalence $\varphi : [U/R] \to \mathcal{F}$ of categories cofibered in groupoids over $\mathcal{C}$.

The following two general lemmas will be used to get presentations.

Lemma 88.25.2. Let $\mathcal{F}$ be category cofibered in groupoids over a category $\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}$. Define $R, s, t, c$ as follows:

  1. $R : \mathcal{C} \to \textit{Sets}$ is the functor $U \times _{f, \mathcal{F}, f} U$.

  2. $t, s : R \to U$ are the first and second projections, respectively.

  3. $c : R \times _{s, U, t} R \to R$ is the morphism given by projection onto the first and last factors of $U \times _{f, \mathcal{F}, f} U \times _{f, \mathcal{F}, f} U$ under the canonical isomorphism $R \times _{s, U, t} R \to U \times _{f, \mathcal{F}, f} U \times _{f, \mathcal{F}, f} U$.

Then $(U, R, s, t, c)$ is a groupoid in functors on $\mathcal{C}$.

Proof. Omitted. $\square$

Lemma 88.25.3. Let $\mathcal{F}$ be category cofibered in groupoids over a category $\mathcal{C}$. Let $U : \mathcal{C} \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}$. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}$ constructed from $f : U \to \mathcal{F}$ in Lemma 88.25.2. Then there is a natural morphism $[f] : [U/R] \to \mathcal{F}$ such that:

  1. $[f]: [U/R] \to \mathcal{F}$ is fully faithful.

  2. $[f]: [U/R] \to \mathcal{F}$ is an equivalence if and only if $f : U \to \mathcal{F}$ is essentially surjective.

Proof. Omitted. $\square$


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