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  "body": "\n## Statement\n\nLet $G$ be an [[abelian group|071DDFVG4J]] and let $n \\ge 1$.\nLet $K(G, n)$ be an [[Eilenberg–MacLane space|AJHEYJE5XY]] that is a CW complex.\nBy the Hurewicz theorem and the universal coefficient theorem, $H^n(K(G, n); G) \\cong \\Hom(\\pi_n K(G, n), G) = \\Hom(G, G)$; let $\\iota \\in H^n(K(G, n); G)$ be the class that corresponds to the identity of $G$.\nLet $X$ be a connected [[CW complex|5W95QFHXPR]] with a $0$-cell as base point.\nLet $[X, K(G, n)]_*$ be the set of base-point-preserving [[homotopy classes|F2F143KQ2M]] of base-point-preserving maps $X \\to K(G, n)$.\nThen the map\n$$[X, K(G, n)]_* \\to H^n(X; G), \\qquad [f] \\mapsto f^*(\\iota),$$\nis a bijection, natural in $X$, onto the [[singular cohomology|6N7BC70CX0]] group $H^n(X; G)$.\n\n## Remarks\n\nOne proof uses obstruction theory.\nA map on the $n$-skeleton that extends over the $(n+1)$-skeleton is given, up to homotopy, by a cellular cocycle.\nThe obstructions to extending it over higher cells lie in groups $H^{k+1}(X; \\pi_k K(G, n))$ with $k \u003e n$, which vanish.\nAnother proof checks that $X \\mapsto H^n(X; G)$ satisfies the hypotheses of the Brown representability theorem and identifies the representing space.\n\nExample: $K(\\Z, 2) = \\C P^\\infty$, so $H^2(X; \\Z)$ is the set of homotopy classes of maps $X \\to \\C P^\\infty$.\nFor $X$ a CW complex, this set also classifies complex line bundles on $X$, and the bijection sends a line bundle to its first Chern class.\n\nThe CW hypothesis on $X$ matters: for spaces that are not of the homotopy type of a CW complex, singular cohomology need not be represented in this way.\nFor example, the Warsaw circle has trivial singular cohomology in positive degrees but nontrivial Čech cohomology, and maps into $K(\\Z, 1) = S^1$ see the Čech groups.\n\n## Explanations\n\n### L1 human\n\nThe $n$-th cohomology of a space with coefficients in $G$ is the same as the set of ways to map the space into one special space $K(G, n)$, up to deformation.\nSo cohomology classes are maps, and computing cohomology is the same as classifying maps into these spaces.\n\n### L2 human\n\nThis theorem turns cohomology into a homotopy-theoretic object.\nA natural operation $H^n(-; G) \\to H^m(-; G')$ then corresponds, by the Yoneda lemma, to a class in $H^m(K(G, n); G')$, so cohomology operations are computed from the cohomology of Eilenberg–MacLane spaces.\nThe spaces $K(G, n)$ form an $\\Omega$-spectrum, since $\\Omega K(G, n+1) \\simeq K(G, n)$, and this spectrum represents ordinary cohomology in the stable setting.\n",
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