cofiber

A cofiber is an algebraic or topological construction dual to a fiber, used primarily in homotopy theory, homological algebra, and stable categories. Given a map \(f:X\to Y\) between topological spaces, the mapping cone \(C_f\) is defined as the quotient of the disjoint union of \(Y\) and the cylinder \(X\times I\) by identifying each \(x\in X\) with \((x,0)\in X\times I\) and each \((x,1)\) with \(f(x)\). The reduced mapping cone, obtained by collapsing the basepoint, is a model for the cofiber. In algebraic topology, the cofiber fits into the long exact sequence of homotopy groups \(\ldots\to\pi_n(X)\to\pi_n(Y)\to\pi_n(C_f)\to\pi_{n-1}(X)\to\ldots\), showing that the cofiber measures the homotopical failure of \(f\) to be a weak equivalence.

In homological algebra, for a chain map \(f:C_\bullet\to D_\bullet\), the mapping cone is the chain complex whose \(n\)-th term is \(D_n\oplus C_{n-1}\) with differential combining those of \(C_\bullet\) and \(D_\bullet\) and \(f\). Its homology yields the derived functor of cokernels and fits into a short exact sequence of chain complexes. The cone construction provides a functorial way to form cofiber sequences, which in a triangulated category correspond to exact triangles \(X\xrightarrow{f}Y\to\operatorname{Cone}(f)\to X[1]\).

In stable homotopy theory, cofibers are generalised to pointed \(\infty\)-categories and spectra. Every map has a cofiber in a stable \(\infty\)-category, and cofiber sequences form the basic building blocks of its tensor triangulated structure. The dual notion of fiber is a limit, whereas the cofiber is a colimit; both are related by the suspension and loop functors. Coefficientwise, the cofiber construction underlies many computational tools, including spectral sequence collapses and Adams spectral sequence differentials. The concept is instrumental in both classical and modern settings, offering a unifying language for excision, killing homotopy elements, and forming pushout squares.