---
tag: 5ZA2T4C3N2
kind: definition
name: Fibration and cofibration
msc: [18N40, 55U35]
defines: HSV1J7BJXX
uses: [SDTZVZ6B4Q, E2870TEDG7, AZK4RD8XV0, PB9JXQ2KC3, Y773B7EQA7, W5WS5EY8ZM]
sources:
  - url: https://en.wikipedia.org/wiki/Model_category
    locator: Definition
---

## Definition

Let $\mathcal{M}$ be a [[model category|SDTZVZ6B4Q]] with [[weak equivalences|E2870TEDG7]] $W$.
A *fibration* is a morphism in the class $\mathrm{Fib}$ of the model structure, and a *cofibration* is a morphism in the class $\mathrm{Cof}$.
A *trivial* (or *acyclic*) fibration is a fibration that is also in $W$, and a trivial cofibration is a cofibration that is also in $W$.
An object $X$ is *fibrant* if the morphism $X \to \ast$ to the terminal object is a fibration.
It is *cofibrant* if the morphism $\emptyset \to X$ from the initial object is a cofibration.

## Remarks

The axioms of a model category imply characterizations by the [[lifting property|AZK4RD8XV0]].
A morphism is a cofibration if and only if it has the left lifting property with respect to all trivial fibrations.
A morphism is a fibration if and only if it has the right lifting property with respect to all trivial cofibrations.
Dually, the trivial cofibrations are the maps with the left lifting property with respect to fibrations.
The trivial fibrations are the maps with the right lifting property with respect to cofibrations.
So, given $W$, the fibrations determine the cofibrations and conversely.

Example: in the Quillen model structure on topological spaces, the fibrations are the [[Serre fibrations|PB9JXQ2KC3]], and every space is fibrant.
In the Strøm model structure on topological spaces, the fibrations are the [[Hurewicz fibrations|Y773B7EQA7]] and the cofibrations are the closed [[cofibrations|W5WS5EY8ZM]] in the sense of the homotopy extension property.
In the projective model structure on non-negatively graded chain complexes of modules over a ring, the fibrations are the maps that are surjective in each positive degree.

Non-example: in the Quillen model structure, the inclusion $p \colon \set{0} \to [0, 1]$ is not a fibration.
The path $t \mapsto t$ in $[0, 1]$ starts at the image $0$ of $p$, but it has no lift to $\set{0}$, because it leaves the image of $p$.
So $p$ fails the right lifting property against the trivial cofibration $\set{0} \to [0, 1]$.