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  "body": "\n## Summary\n\nSingular cohomology assigns to a topological space $X$ and an abelian group $G$ the groups $H^n(X; G)$.\nThey are built from functions on singular simplices with values in $G$.\nIt is the dual of [[singular homology|M0Q5CFJ7ZH]], and it turns continuous maps into homomorphisms in the opposite direction.\nIts extra structure, the cup product, makes $H^\\bullet(X; R)$ a ring, which distinguishes spaces that have the same homology groups.\n\n## Explanations\n\n### L1 human\n\nHomology builds things out of simplices; cohomology measures them.\nA cochain is a rule that gives a number to every simplex, like a test that is applied to each piece of the space.\nA cocycle is a test that gives zero on every boundary, so its value on a cycle depends only on the homology class.\nFor example, on a circle the rule \"count how many times a loop winds around\" is a $1$-cocycle, and it generates $H^1(S^1; \\Z) \\cong \\Z$.\n",
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