Cofiibrations

Cofibrations are a fundamental concept in homotopy theory, a branch of mathematics that studies topological spaces and maps between them up to continuous deformation. Informally, a cofibration is a type of map that behaves nicely with respect to attaching new pieces to a space. More precisely, a map $i: X \to Y$ is a cofibration if it satisfies the homotopy extension property. This means that if we have a space $Z$ and a map $f: Y \to Z$ and a homotopy $h: X \times I \to Z$ such that $h(x, 0) = f(i(x))$ for all $x \in X$, then there exists a homotopy $H: Y \times I \to Z$ such that $H(y, 0) = f(y)$ for all $y \in Y$ and $H(i(x), t) = h(x, t)$ for all $x \in X$ and $t \in I$. Here, $I$ denotes the unit interval $[0, 1]$.

This property essentially states that any homotopy starting on the image of $X$ under $i$ can be extended to a homotopy on all of $Y$. This is crucial for constructing and understanding spaces built by attaching smaller spaces, such as skeleta of CW complexes. For example, inclusion maps of cells into a CW complex are cofibrations. Cofibrations are dual to fibrations, which satisfy a homotopy lifting property. The study of cofibrations is intimately linked with the theory of homotopy pushouts and the construction of topological spaces.