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  "body": "\n## Definition\n\nLet $q \\ge 0$ be an integer.\nLet $\\Gamma_q$ be the [[groupoid|FH3C0T643F]] whose objects are the points of $\\R^q$.\nIts arrows from $a$ to $b$ are the [[germs|VZXF7RNFKS]] at $a$ of local diffeomorphisms $\\phi$ of $\\R^q$ with $\\phi(a) = b$.\nGive the set of arrows the sheaf topology, in which the germs at nearby points of one local diffeomorphism form an open set.\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA *$\\Gamma_q$-cocycle* on $X$ is given by an open cover $(U_i)_{i \\in I}$ of $X$ and continuous maps $\\gamma_{ij} \\colon U_i \\cap U_j \\to \\Gamma_q$ such that\n$$\\gamma_{ik}(x) = \\gamma_{ij}(x) \\, \\gamma_{jk}(x) \\quad \\text{for all } x \\in U_i \\cap U_j \\cap U_k.$$\nTwo $\\Gamma_q$-cocycles are equivalent if their union, over the disjoint union of the two index sets, is again a $\\Gamma_q$-cocycle.\nA *Haefliger structure* of codimension $q$ on $X$, or $\\Gamma_q$-structure, is an equivalence class of $\\Gamma_q$-cocycles.\n\n## Remarks\n\nThe cocycle condition with $i = j = k$ shows that $\\gamma_{ii}(x)$ is the germ of the identity at a point $f_i(x) \\in \\R^q$, and this defines continuous maps $f_i \\colon U_i \\to \\R^q$.\nThe germ $\\gamma_{ij}(x)$ goes from $f_j(x)$ to $f_i(x)$, so near $x$ the local maps satisfy $f_i = \\phi \\circ f_j$ for a representative $\\phi$.\nThe same definition works with any topological groupoid in place of $\\Gamma_q$.\n\nExample: let $\\mathcal{F}$ be a [[foliation|98HTB7MSPX]] of codimension $q$ of a manifold $X$, given by submersions $f_i \\colon U_i \\to \\R^q$ with $f_i = g_{ij} \\circ f_j$ on overlaps for local diffeomorphisms $g_{ij}$.\nTaking $\\gamma_{ij}(x)$ to be the germ of $g_{ij}$ at $f_j(x)$ gives a $\\Gamma_q$-cocycle, since the transition maps compose as $g_{ik} = g_{ij} \\circ g_{jk}$ near each point.\nAnother example: any continuous map $f \\colon X \\to \\R^q$ gives a $\\Gamma_q$-structure with one open set $U_1 = X$ and $\\gamma_{11}(x)$ the germ of the identity at $f(x)$.\n\nNon-example: let $X$ be a point, $U_1 = U_2 = X$, $\\gamma_{11} = \\gamma_{22} = \\gamma_{21}$ the germ of the identity at $0$ and $\\gamma_{12}$ the germ of $t \\mapsto 2t$ at $0$.\nThe cocycle condition fails for $(i, j, k) = (1, 2, 1)$, since $\\gamma_{12} \\gamma_{21}$ is the germ of $t \\mapsto 2t$ and not that of the identity.\n\nHaefliger structures pull back along continuous maps.\nFor a CW complex $X$, homotopy classes of $\\Gamma_q$-structures on $X$ correspond to homotopy classes of maps from $X$ into a [[classifying space|WSZ40H5E4T]] $B\\Gamma_q$.\n\n## Explanations\n\n### L1 human\n\nA Haefliger structure is the gluing data of a foliation, kept even where the local maps fail to be submersions.\n",
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