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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[abelian groups|082GHNPJ53]] and let $n, m \\ge 0$ be integers.\nWrite $H^n(X; G)$ for [[singular cohomology|6N7BC70CX0]] with coefficients in $G$.\nA *cohomology operation* of type $(G, n; H, m)$ is a family of functions $\\theta_X \\colon H^n(X; G) \\to H^m(X; H)$, one for each topological space $X$.\nThe family must be natural.\nThat is, $f^* \\circ \\theta_X = \\theta_Y \\circ f^*$ for every [[continuous map|0X3WGX95FE]] $f \\colon Y \\to X$.\nHere $f^*$ denotes the induced maps in cohomology.\n\n## Remarks\n\nIn the language of categories, a cohomology operation is a natural transformation between the functors $H^n(-; G)$ and $H^m(-; H)$ to sets.\nThe functions $\\theta_X$ need not be homomorphisms.\nOperations of type $(G, n; H, m)$ correspond bijectively to elements of $H^m(K(G, n); H)$, through $\\theta \\mapsto \\theta(\\iota_n)$, where $\\iota_n \\in H^n(K(G, n); G)$ is the fundamental class of the Eilenberg–MacLane space.\n\nExample: the cup square $\\theta_X(x) = x \\smile x$ is an operation of type $(\\Z, n; \\Z, 2n)$.\nThe check: $f^*(x \\smile x) = f^* x \\smile f^* x$, because $f^*$ is a ring homomorphism.\nIt is not additive in general: $(x + y)^2 = x^2 + x y + y x + y^2$.\n\nNon-example: on $H^0(-; \\Z)$, let $\\theta_X(x) = x$ if $X$ is path-connected and $\\theta_X(x) = 0$ otherwise.\nTake $X = \\set{a, b}$ discrete, $Y$ a point, $f$ the map to $a$, and $x = (1, 1) \\in H^0(X; \\Z) \\cong \\Z^2$.\nThen $f^* \\theta_X(x) = 0$ but $\\theta_Y(f^* x) = 1$, so the naturality condition fails.\n",
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