Cofibrations

Cofibrations are a fundamental concept in homotopy theory, a branch of mathematics that studies topological spaces and their continuous maps up to a certain equivalence relation. They were introduced by J. H. C. Whitehead in the 1950s as a way to generalize the notion of a subspace inclusion.

A map f: A → X in a topological space is called a cofibration if it has the left lifting property with respect to all fibrations. This means that for any commutative diagram of the form

A → X

↓ ↓

Y → Z

where the vertical map is a fibration, there exists a dashed arrow A → Y making the diagram commute. Intuitively, this means that the map f can be "extended" along any fibration.

Cofibrations are closely related to fibrations, which are maps that have the right lifting property with respect to all cofibrations. Together, they form a dual pair of notions that play a crucial role in various areas of mathematics, including algebraic topology, differential topology, and homotopy theory itself.

One of the most important properties of cofibrations is that they preserve homotopy groups. This means that if f: A → X is a cofibration, then the induced map f_*: π_n(A) → π_n(X) on homotopy groups is injective for all n ≥ 1. This property makes cofibrations a useful tool for studying the homotopy type of topological spaces.

In recent years, the theory of cofibrations has been further developed and generalized to more abstract settings, such as model categories and simplicial sets. These generalizations have led to new insights and applications in various areas of mathematics.