subfibration

A subfibration is a concept in mathematics, specifically within the field of algebraic topology, that refers to a specific type of map between fibrations. A fibration is a continuous map between two fiber bundles that has certain lifting properties, and subfibrations are used to describe a particular relationship between two such fibrations.

In more detail, given two fibrations \( p: E \rightarrow B \) and \( q: E' \rightarrow B' \), a subfibration is a map \( f: E \rightarrow E' \) that respects the fibration structures. This means that \( f \) maps fibers of \( p \) to fibers of \( q \) in a way that is compatible with the projection maps \( p \) and \( q \). Essentially, a subfibration is a map between the total spaces of two fibrations that preserves the fibration structure.

The concept of a subfibration is important in the study of fiber bundles and fibrations, as it allows for the comparison and manipulation of different fibrations in a structured manner. It is often used in the context of homotopy theory, where understanding the relationships between different fibrations can provide insights into the topological properties of the spaces involved.

Subfibrations can be seen as a generalization of the notion of a subbundle in the context of fiber bundles. Just as a subbundle is a fiber bundle that is contained within another fiber bundle, a subfibration is a fibration that is contained within another fibration in a specific sense. This containment is defined by the map \( f \) that respects the fibration structures.

In summary, a subfibration is a map between two fibrations that preserves the fibration structure, allowing for the comparison and manipulation of different fibrations in algebraic topology.