{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "6R57T5W7BV"
    }
  ],
  "body": "\n## Description\n\nLet $G$ be an [[abelian group|071DDFVG4J]].\nThe Eilenberg–MacLane spectrum $HG$ is the [[spectrum|1GHF4EKBXS]] with $(HG)_n = K(G, n)$, an [[Eilenberg–MacLane space|AJHEYJE5XY]], for $n \\ge 0$.\nIts structure maps $\\Sigma K(G, n) \\to K(G, n+1)$ are adjoint to weak homotopy equivalences $K(G, n) \\to \\Omega K(G, n+1)$, so $HG$ is an $\\Omega$-spectrum.\nIts homotopy groups are $\\pi_0(HG) = G$ and $\\pi_k(HG) = 0$ for $k \\neq 0$, and these properties determine $HG$ up to isomorphism in the [[stable homotopy category|26GTNMAF7B]].\n\n## Remarks\n\nThe spectrum $HG$ represents ordinary [[singular cohomology|6N7BC70CX0]] with coefficients in $G$: for a CW complex $X$ and $n \\ge 0$, $\\tilde H^n(X; G) \\cong [\\Sigma^\\infty X, \\Sigma^n HG]$.\nIt also represents homology: $\\tilde H_n(X; G) \\cong \\pi_n(HG \\wedge X)$.\nFor $p$ prime, the graded ring $[H\\F_p, \\Sigma^* H\\F_p]$ of stable self-maps is the mod $p$ Steenrod algebra.\nIf $R$ is a commutative ring, the product of $R$ makes $HR$ a commutative ring spectrum.\n\n## Explanations\n\n### L1 human\n\nThe Eilenberg–MacLane spectrum of a group $G$ is the simplest spectrum with homotopy $G$: it has $G$ in degree zero and nothing else.\nIt is the stable object behind ordinary cohomology with coefficients in $G$.\nComparing other spectra with it is how one connects generalized cohomology theories with ordinary cohomology.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "instance_of": [
      "1GHF4EKBXS"
    ],
    "kind": "object",
    "msc": [
      "55P42",
      "55P20"
    ],
    "name": "Eilenberg–MacLane spectrum",
    "same_as": {
      "wikidata": "Q134310487",
      "wikipedia": "Eilenberg–MacLane spectrum"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_spectrum"
      }
    ],
    "tag": "XJPYJT9Z1P",
    "uses": [
      "1GHF4EKBXS",
      "AJHEYJE5XY",
      "071DDFVG4J",
      "6N7BC70CX0",
      "26GTNMAF7B"
    ]
  },
  "kind": "object",
  "link": "https://math2.tamnd.com/node/XJPYJT9Z1P/",
  "links": [
    {
      "field": "instance_of",
      "target": "1GHF4EKBXS"
    },
    {
      "field": "uses",
      "target": "1GHF4EKBXS"
    },
    {
      "field": "uses",
      "target": "AJHEYJE5XY"
    },
    {
      "field": "uses",
      "target": "071DDFVG4J"
    },
    {
      "field": "uses",
      "target": "6N7BC70CX0"
    },
    {
      "field": "uses",
      "target": "26GTNMAF7B"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/XJPYJT9Z1P.md",
  "msc": [
    "55P42",
    "55P20"
  ],
  "name": "Eilenberg–MacLane spectrum",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XJP/XJPYJT9Z1P.md",
  "status": null,
  "tag": "XJPYJT9Z1P",
  "uri": "m2://XJPYJT9Z1P",
  "url": "https://math2.tamnd.com/wiki/eilenberg-maclane-spectrum/",
  "writing": {
    "tag": "XJPYJT9Z1P",
    "kind": "object",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}