Eckmann–Hilton duality

Definition

Eckmann–Hilton duality is an informal correspondence between notions of homotopy theory of pointed spaces. It reverses the direction of all maps and exchanges the members of each of the following pairs:

1. maps out of the cylinder $X \times [0, 1]$, and maps into the path space $Y^{[0, 1]}$, as two descriptions of a homotopy;
2. the reduced suspension $\Sigma X$, and the loop space $\Omega Y$;
3. cofibrations, defined by extending homotopies, and fibrations, defined by lifting homotopies;
4. the wedge sum $X \vee Y$, and the product $X \times Y$;
5. mapping cones and cofibre sequences, and homotopy fibres and fibre sequences;
6. the homotopy groups $\pi_n(Y) = [S^n, Y]_*$, and the cohomology groups $H^n(X; G) \cong [X, K(G, n)]$ of a CW complex $X$, with $K(G, n)$ an Eilenberg–MacLane space.