Cofibration

In mathematics, in particular homotopy theory, a continuous mapping between topological spaces

${\displaystyle i:A\to X}$,

is a cofibration if it has the homotopy extension property with respect to all topological spaces ${\displaystyle S}$. That is, ${\displaystyle i}$ is a cofibration if for each topological space ${\displaystyle S}$, and for any continuous maps ${\displaystyle f,f':A\to S}$ and ${\displaystyle g:X\to S}$ with ${\displaystyle g\circ i=f}$, for any homotopy ${\displaystyle h:A\times I\to S}$ from ${\displaystyle f}$ to ${\displaystyle f'}$, there is a continuous map ${\displaystyle g':X\to S}$ and a homotopy ${\displaystyle h':X\times I\to S}$ from ${\displaystyle g}$ to ${\displaystyle g'}$ such that ${\displaystyle h'(i(a),t)=h(a,t)}$ for all ${\displaystyle a\in A}$ and ${\displaystyle t\in I}$. (Here, ${\displaystyle I}$ denotes the unit interval ${\displaystyle [0,1]}$.)

This definition is formally dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton duality in topology.

Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of model category as a formal framework for doing homotopy theory in more general categories; a model category is endowed with three distinguished classes of morphisms called fibrations, cofibrations and weak equivalences satisfying certain lifting and factorization axioms.