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  "body": "\n## Definition\n\n*Eckmann–Hilton duality* is an informal correspondence between notions of homotopy theory of [[pointed spaces|T9DG8DDE75]].\nIt reverses the direction of all maps and exchanges the members of each of the following pairs:\n\n1. maps out of the cylinder $X \\times [0, 1]$, and maps into the path space $Y^{[0, 1]}$, as two descriptions of a homotopy;\n2. the reduced [[suspension|Q7RFNJH76Q]] $\\Sigma X$, and the [[loop space|2JFF5F3EGF]] $\\Omega Y$;\n3. [[cofibrations|W5WS5EY8ZM]], defined by extending homotopies, and fibrations, defined by lifting homotopies;\n4. the [[wedge sum|5JSWM1HBX2]] $X \\vee Y$, and the product $X \\times Y$;\n5. mapping cones and cofibre sequences, and homotopy fibres and fibre sequences;\n6. 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.\n\n## Remarks\n\nThe pair in clause 2 is linked by the natural bijection $[\\Sigma X, Y]_* \\cong [X, \\Omega Y]_*$.\nExample: the homotopy extension property of a cofibration corresponds under clause 3 to the homotopy lifting property of a fibration.\nLikewise the cofibre sequence of a map corresponds to its fibre sequence.\nThe correspondence is not a theorem: the dual of a true statement need not be true, and each dual statement needs its own proof.\nThe duality was named after Beno Eckmann and Peter Hilton.\n",
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