Cofibration

A cofibration is a concept in abstract mathematics, specifically within the field of category theory. It is a type of morphism, or arrow, between objects in a category. A morphism $f: X \to Y$ is called a cofibration if it satisfies a certain lifting property. This property states that for any commutative diagram involving $X$ and $Y$ and a "small" object $I$ with a map $g: I \to Y$, there exists a map $h: I \to X$ such that $f \circ h = g$. The "small" object $I$ is typically defined relative to a specific class of objects and morphisms in the category.

This lifting property is dual to the concept of a fibration. In essence, fibrations deal with how objects "over" a base object are structured, while cofiibrations concern how objects "under" a base object can be constructed. Cofiibrations are fundamental in the study of model categories, which provide a framework for defining and working with notions of approximation and homology in a very general setting. They play a crucial role in defining the class of "cofibrant objects" within a model category, which are objects that behave well with respect to certain constructions and constructions. The existence and properties of cofiibrations are central to many advanced topics in algebraic topology and category theory.