Subfibrations

Subfibrations are a generalization of fibrations in the context of category theory, which was introduced by Peter May in the 1970s. A subfibration is a subobject of a fibration, where a subobject is a morphism from an object to its subobject. In the context of a fibration, a subfibration is a morphism with certain properties that are preserved by the pullback operation.

To be more specific, given a fibration p: E -> B with a subobject f: A -> E, we can form the pullback A x_B E. This pullback is again a full subobject of E that is closed under composition, and it satisfies the universal property of the pullback. The pair (A, f) is then a subfibration of the fibration p, which we denote as A -> B.

The concept of subfibrations has important applications in various areas of mathematics, including homotopy theory, algebraic topology, and homological algebra. Subfibrations play a crucial role in the theory of model categories, where they help in constructing and manipulating fibrant objects. They are also used in the context of simplicial and cosimplicial spaces.

One of the key properties of subfibrations is their behavior under composition and pullback operations. Subfibrations are closed under composition, meaning that the composition of two subfibrations is again a subfibration. Moreover, they are preserved by the pullback operation, meaning that the pullback of a subfibration is again a subfibration.

Subfibrations have been studied extensively in the literature, and their properties and applications have been extensively investigated. They are a fundamental concept in category theory and have far-reaching implications in various areas of mathematics.