category theory

# Contents

## Idea

The notion of cofibration is dual to that of fibration. See there for more details.

A cofibration is a member of a distinguished class of cofibrations in one of the several setups in homotopy theory:

In traditional topology, one usually means a Hurewicz cofibration.

In category theory and descent theory, Grothendieck however introduced a notion of cofibered category or cofibration, whose definition is categorically dual to that of a fibered category and based on the generalization of the universal property of coCartesian squares; however it does not correspond to the extension property in topology, but to a lifting property, like for fibrations. For that reason, Gray suggested (and this is to some extent adopted in $n$lab) to call such categories opfibrations. In the quasicategorical setup, the generalizations of fibered categories are called Cartesian fibrations, and the generalizations of op/co-fibered categories are called coCartesian fibrations. In the $1$-categorical setup, the coCartesian arrows in op/co-fibrations are indeed the ones which complete coCartesian squares in the special and most important case of domain opfibrations.

## Stability properties

Cofibrations are usually defined in such a way that they are stable at least under the following operations in the category under consideration

• composition
• pushouts of spans at least one of whose legs is a cofibration

(Please mind the precise definitions of the category you are using. Also compare the stability properties of the dual notion fibration.)

Revised on May 31, 2017 15:05:21 by Peter Heinig (84.183.83.131)