Definition 90.5.1. 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 $p^{opp}: \mathcal{F}^{opp} \to \mathcal{C}^{opp}$.

## 90.5 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.

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

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 $f$.

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 90.5.2. 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.

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

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 $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{F}(U)}(x, y)$.

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 $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{F}(U)}(x, x)$.

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 $f$.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.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.

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 $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(\mathcal{F}, \mathcal{F}')$.

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 $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F})$.

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.

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.

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.

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.

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.

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.

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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