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  "body": "\n## Definition\n\nLet $G$ be a discrete group.\nThe *orbit category* $\\mathcal{O}_G$ has as objects the left $G$-sets $G/H$ for subgroups $H \\le G$, and as morphisms the $G$-equivariant maps.\nA *coefficient system* is a contravariant functor $M \\colon \\mathcal{O}_G \\to \\mathrm{Ab}$ to [[abelian groups|082GHNPJ53]].\n\nLet $X$ be a $G$-CW complex, that is, a [[CW complex|5W95QFHXPR]] with a cellular [[action|0WCY42EV9V]] of $G$.\nIt is required that each element of $G$ that maps a cell to itself fixes that cell pointwise.\nFor each subgroup $H$, the fixed set $X^H$ is a subcomplex.\nFor $n \\ge 0$, let $\\underline{C}_n(X)$ be the contravariant functor $\\mathcal{O}_G \\to \\mathrm{Ab}$ given by $G/H \\mapsto C_n(X^H)$, the [[cellular chains|W4DG4WCQ5W]] of $X^H$.\nA $G$-map $G/H \\to G/K$ with $H \\mapsto gK$ acts by $C_n(X^K) \\to C_n(X^H)$, $c \\mapsto g c$.\nThe cellular boundary maps make $\\underline{C}_\\bullet(X)$ a chain complex of functors.\nThe *Bredon cochain complex* is\n$$C^n_G(X; M) = \\operatorname{Nat}(\\underline{C}_n(X), M),$$\nthe group of natural transformations, with coboundary induced by the boundary of $\\underline{C}_\\bullet(X)$.\nThe *Bredon cohomology* $H^n_G(X; M)$ is the $n$-th cohomology of $C^\\bullet_G(X; M)$.\n\n## Remarks\n\nExample: let $X$ be a point with the trivial action.\nThen $X^H$ is a point for every $H$, so $\\underline{C}_0(X)$ is the constant functor $\\Z$ and $\\underline{C}_n(X) = 0$ for $n \u003e 0$.\nA natural transformation from the constant functor $\\Z$ to $M$ is fixed by its value at $G/G$, the terminal object of $\\mathcal{O}_G$.\nSo $H^0_G(\\mathrm{pt}; M) \\cong M(G/G)$ and $H^n_G(\\mathrm{pt}; M) = 0$ for $n \u003e 0$.\n\nIf $G$ acts freely on $X$ and $M$ is the constant functor with value $A$, then $X^H$ is empty for $H \\neq 1$ and $H^n_G(X; M) \\cong H^n(X/G; A)$.\n\nNon-example: a family of groups $M(G/H)$ with maps going in the covariant direction is not a coefficient system for this cohomology.\nCovariant functors on $\\mathcal{O}_G$ are the coefficients of Bredon homology instead.\n",
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