---
tag: H4NTJ12J88
kind: definition
name: Fibred category
msc: [18D30]
defines: NT6WDN07RC
uses: [28NQ0KRHD3, Q05K04QJ71]
sources:
  - url: https://en.wikipedia.org/wiki/Fibred_category
---

## Definition

Let $p \colon \mathcal{E} \to \mathcal{B}$ be a [[functor|28NQ0KRHD3]].
A morphism $\varphi \colon x \to y$ of $\mathcal{E}$ is *cartesian* if this holds.
For every morphism $\psi \colon z \to y$ of $\mathcal{E}$ and every morphism $g \colon p(z) \to p(x)$ of $\mathcal{B}$ with $p(\varphi) \circ g = p(\psi)$, there is a unique $\chi \colon z \to x$ with $\varphi \circ \chi = \psi$ and $p(\chi) = g$.
The functor $p$ is a *fibration*, and $\mathcal{E}$ is *fibred* over $\mathcal{B}$, if this holds.
For every object $y$ of $\mathcal{E}$ and every morphism $f \colon b \to p(y)$ of $\mathcal{B}$, there is a cartesian morphism $\varphi \colon x \to y$ with $p(\varphi) = f$.

## Remarks

The cartesian morphism $\varphi$ over $f$ is unique up to a unique isomorphism over $\id_b$, and $x$ is often written $f^* y$.
The fibre over $b$ is the subcategory of objects over $b$ and morphisms over $\id_b$.
Fibrations over $\mathcal{B}$ correspond to pseudofunctors from $\mathcal{B}^{\mathrm{op}}$ to categories, by the Grothendieck construction.

Example: let $\mathcal{B}$ have [[pullbacks|Q05K04QJ71]], let $\mathcal{E}$ be the arrow category whose objects are morphisms $e \colon a \to b$ of $\mathcal{B}$, and let $p$ send $e$ to its codomain $b$.
Given $e \colon a \to b$ and $f \colon b' \to b$, the pullback square of $e$ along $f$ is a morphism of $\mathcal{E}$ over $f$.
The universal property of the pullback says exactly that it is cartesian, so $p$ is a fibration.

Non-example: let $\mathcal{B}$ be the category with objects $0, 1$ and one non-identity arrow $f \colon 0 \to 1$.
Let $\mathcal{E}$ have one object $y$ and only its identity, and let $p(y) = 1$.
No object of $\mathcal{E}$ lies over $0$, so there is no morphism over $f$ ending at $y$, and $p$ is not a fibration.