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  "body": "\n## Summary\n\nA Haefliger structure is a generalisation of a [[foliation|98HTB7MSPX]] in which the local defining maps need not be submersions.\nIt is given by local maps to $\\R^q$ glued by [[germs|VZXF7RNFKS]] of local diffeomorphisms.\nIn other words, it is a cocycle with values in the groupoid of such germs.\nUnlike foliations, Haefliger structures can be pulled back along any continuous map.\nUp to homotopy, they are classified by maps into a [[classifying space|WSZ40H5E4T]] $B\\Gamma_q$.\nThis homotopy theory is the basis of the existence theory of foliations.\n\n## Explanations\n\n### L1 human\n\nA foliation of codimension $q$ is locally described by a submersion to $\\R^q$, whose level sets are the pieces of leaves.\nOn overlaps the local submersions differ by local diffeomorphisms of $\\R^q$.\nA Haefliger structure keeps the gluing data but drops the requirement that the local maps be submersions.\nA constant map to $\\R^q$ already gives one, even though it does not foliate anything.\nThis flexibility makes Haefliger structures behave like vector bundles: they pull back along maps and have a classifying space.\n",
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