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  "body": "\n## Summary\n\nThe bar resolution of a group $G$ is an explicit free resolution of the trivial module $\\Z$ over the [[group ring|MNP2HBWB5Q]] $\\Z[G]$.\nIts basis elements in degree $n$ are the symbols $[g_1 | \\cdots | g_n]$, one for each list of $n$ elements of $G$.\nIt turns [[group cohomology|YXJXA0Q52R]] into a concrete computation with functions of $n$ group elements, which gives the classical cocycle descriptions of $H^1$ and $H^2$.\nThe same construction exists for augmented algebras, where it computes Hochschild homology and Tor.\n\n## Explanations\n\n### L1 human\n\nTo compute group cohomology one needs a resolution, and the bar resolution is one that works for every group without any choices.\nIts pieces are free modules with a basis vector for every list of group elements.\nThe boundary map multiplies neighbouring entries of the list, with alternating signs, and drops the first or last entry at the ends.\nIt is large and wasteful for computing, but its uniformity makes it the right tool for proofs and for writing down cocycles.\n",
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