## 88.5 Categories cofibered in groupoids

In developing the theory we work with categories cofibered in groupoids. We assume as known the definition and basic properties of categories fibered in groupoids, see Categories, Section 4.35.

Definition 88.5.1. Let $\mathcal{C}$ be a category. A category cofibered in groupoids over $\mathcal{C}$ is a category $\mathcal{F}$ equipped with a functor $p: \mathcal{F} \to \mathcal{C}$ such that $\mathcal{F}^{opp}$ is a category fibered in groupoids over $\mathcal{C}^{opp}$ via $p^{opp}: \mathcal{F}^{opp} \to \mathcal{C}^{opp}$.

Explicitly, $p: \mathcal{F} \to \mathcal{C}$ is cofibered in groupoids if the following two conditions hold:

1. For every morphism $f: U \to V$ in $\mathcal{C}$ and every object $x$ lying over $U$, there is a morphism $x \to y$ of $\mathcal{F}$ lying over $f$.

2. For every pair of morphisms $a: x \to y$ and $b: x \to z$ of $\mathcal{F}$ and any morphism $f: p(y) \to p(z)$ such that $p(b) = f \circ p(a)$, there exists a unique morphism $c: y \to z$ of $\mathcal F$ lying over $f$ such that $b = c \circ a$.

Remarks 88.5.2. Everything about categories fibered in groupoids translates directly to the cofibered setting. The following remarks are meant to fix notation. Let $\mathcal{C}$ be a category.

1. We often omit the functor $p: \mathcal{F} \to \mathcal{C}$ from the notation.

2. The fiber category over an object $U$ in $\mathcal{C}$ is denoted by $\mathcal{F}(U)$. Its objects are those of $\mathcal{F}$ lying over $U$ and its morphisms are those of $\mathcal{F}$ lying over $\text{id}_ U$. If $x, y$ are objects of $\mathcal{F}(U)$, we sometimes write $\mathop{\mathrm{Mor}}\nolimits _ U(x, y)$ for $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{F}(U)}(x, y)$.

3. The fibre categories $\mathcal{F}(U)$ are groupoids, see Categories, Lemma 4.35.2. Hence the morphisms in $\mathcal{F}(U)$ are all isomorphisms. We sometimes write $\text{Aut}_ U(x)$ for $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{F}(U)}(x, x)$.

4. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}$, let $f: U \to V$ be a morphism in $\mathcal{C}$, and let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(U))$. A pushforward of $x$ along $f$ is a morphism $x \to y$ of $\mathcal{F}$ lying over $f$. A pushforward is unique up to unique isomorphism (see the discussion following Categories, Definition 4.33.1). We sometimes write $x \to f_*x$ for “the” pushforward of $x$ along $f$.

5. A choice of pushforwards for $\mathcal{F}$ is the choice of a pushforward of $x$ along $f$ for every pair $(x, f)$ as above. We can make such a choice of pushforwards for $\mathcal{F}$ by the axiom of choice.

6. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}$. Given a choice of pushforwards for $\mathcal{F}$, there is an associated pseudo-functor $\mathcal{C} \to \textit{Groupoids}$. We will never use this construction so we give no details.

7. A morphism of categories cofibered in groupoids over $\mathcal{C}$ is a functor commuting with the projections to $\mathcal{C}$. If $\mathcal{F}$ and $\mathcal{F}'$ are categories cofibered in groupoids over $\mathcal{C}$, we denote the morphisms from $\mathcal{F}$ to $\mathcal{F}'$ by $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(\mathcal{F}, \mathcal{F}')$.

8. Categories cofibered in groupoids form a $(2, 1)$-category $\text{Cof}(\mathcal{C})$. Its 1-morphisms are the morphisms described in (7). If $p : \mathcal{F} \to C$ and $p': \mathcal{F}' \to \mathcal{C}$ are categories cofibered in groupoids and $\varphi , \psi : \mathcal{F} \to \mathcal{F}'$ are $1$-morphisms, then a 2-morphism $t : \varphi \to \psi$ is a morphism of functors such that $p'(t_ x) = \text{id}_{p(x)}$ for all $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F})$.

9. Let $F : \mathcal{C} \to \textit{Groupoids}$ be a functor. There is a category cofibered in groupoids $\mathcal{F} \to \mathcal{C}$ associated to $F$ as follows. An object of $\mathcal{F}$ is a pair $(U, x)$ where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x \in \mathop{\mathrm{Ob}}\nolimits (F(U))$. A morphism $(U, x) \to (V, y)$ is a pair $(f, a)$ where $f \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, V)$ and $a \in \mathop{\mathrm{Mor}}\nolimits _{F(V)}(F(f)(x), y)$. The functor $\mathcal{F} \to \mathcal{C}$ sends $(U, x)$ to $U$. See Categories, Section 4.37.

10. Let $\mathcal{F}$ be cofibered in groupoids over $\mathcal{C}$. For $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ set $\overline{\mathcal{F}}(U)$ equal to the set of isomorphisms classes of the category $\mathcal{F}(U)$. If $f : U \to V$ is a morphism of $\mathcal{C}$, then we obtain a map of sets $\overline{\mathcal{F}}(U) \to \overline{\mathcal{F}}(V)$ by mapping the isomorphism class of $x$ to the isomorphism class of a pushforward $f_*x$ of $x$ see (4). Then $\overline{\mathcal{F}} : \mathcal{C} \to \textit{Sets}$ is a functor. Similarly, if $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of cofibered categories, we denote by $\overline{\varphi }: \overline{\mathcal{F}} \to \overline{\mathcal{G}}$ the associated morphism of functors.

11. Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. We can think of a set as a discrete category, i.e., as a groupoid with only identity morphisms. Then the construction (9) associates to $F$ a category cofibered in sets. This defines a fully faithful embedding of the category of functors $\mathcal{C} \to \textit{Sets}$ to the category of categories cofibered in groupoids over $\mathcal{C}$. We identify the category of functors with its image under this embedding. Hence if $F : \mathcal{C} \to \textit{Sets}$ is a functor, we denote the associated category cofibered in sets also by $F$; and if $\varphi : F \to G$ is a morphism of functors, we denote still by $\varphi$ the corresponding morphism of categories cofibered in sets, and vice-versa. See Categories, Section 4.38.

12. Let $U$ be an object of $\mathcal{C}$. We write $\underline{U}$ for the functor $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, -): \mathcal{C} \to \textit{Sets}$. This defines a fully faithful embedding of $\mathcal C^{opp}$ into the category of functors $\mathcal{C} \to \textit{Sets}$. Hence, if $f : U \to V$ is a morphism, we are justified in denoting still by $f$ the induced morphism $\underline{V} \to \underline{U}$, and vice-versa.

13. Fiber products of categories cofibered in groupoids: If $\mathcal{F} \to \mathcal{H}$ and $\mathcal{G} \to \mathcal{H}$ are morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$, then a construction of their 2-fiber product is given by the construction for their 2-fiber product as categories over $\mathcal{C}_\Lambda$, as described in Categories, Lemma 4.32.3.

14. Products of categories cofibered in groupoids: If $\mathcal{F}$ and $\mathcal{G}$ are categories cofibered in groupoids over $\mathcal{C}_\Lambda$ then their product is defined to be the $2$-fiber product $\mathcal{F} \times _{\mathcal{C}_\Lambda } \mathcal{G}$ as described in Categories, Lemma 4.32.3.

15. Restricting the base category: Let $p : \mathcal{F} \to \mathcal{C}$ be a category cofibered in groupoids, and let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$. The restriction $\mathcal{F}|_{\mathcal{C}'}$ is the full subcategory of $\mathcal{F}$ whose objects lie over objects of $\mathcal{C}'$. It is a category cofibered in groupoids via the functor $p|_{\mathcal{C}'}: \mathcal{F}|_{\mathcal{C}'} \to \mathcal{C}'$.

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