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  "body": "\n## Definition\n\nLet $\\mathcal{M}$ be a [[model category|SDTZVZ6B4Q]] and let $I$ be a small category.\nAssume that the [[functor category|1X0WKFQX8K]] $\\mathcal{M}^I$ carries the injective model structure.\nIn it, [[weak equivalences|E2870TEDG7]] and cofibrations are defined objectwise; it exists, for instance, when $\\mathcal{M}$ is combinatorial.\nThe [[limit|CHS5AFJGWE]] functor $\\lim \\colon \\mathcal{M}^I \\to \\mathcal{M}$ is right [[adjoint|TZHF6ZM92S]] to the constant diagram functor, which preserves cofibrations and trivial cofibrations, so $\\lim$ is a right [[Quillen functor|SY38F954FE]].\nThe *homotopy limit* is its total right derived functor\n$$\\operatorname{holim} = \\mathbf{R}\\lim \\colon \\mathrm{Ho}(\\mathcal{M}^I) \\to \\mathrm{Ho}(\\mathcal{M}),$$\nbetween the [[homotopy categories|48QQ7T98Z9]].\nOn a diagram $D$ it is computed as $\\lim R D$, where $D \\to RD$ is a weak equivalence to a [[fibrant|HSV1J7BJXX]] diagram in the injective model structure.\n\n## Remarks\n\nThe homotopy limit sends objectwise weak equivalences of diagrams to weak equivalences, and it comes with a natural map $\\lim D \\to \\operatorname{holim} D$.\n\nExample: for a diagram $X \\xrightarrow{f} Z \\xleftarrow{g} Y$ of topological spaces, the homotopy limit is the homotopy pullback\n$$X \\times_Z Z^{[0,1]} \\times_Z Y = \\set{(x, \\omega, y) : \\omega \\colon [0, 1] \\to Z,\\ \\omega(0) = f(x),\\ \\omega(1) = g(y)}.$$\nWhen $f$ or $g$ is a Serre fibration, the map from the ordinary pullback to the homotopy pullback is a weak equivalence.\n\nNon-example: the ordinary limit is not homotopy invariant.\nThere is a map from the diagram $\\set{0} \\to \\R \\leftarrow \\set{1}$ to the diagram $\\set{0} \\to \\R \\leftarrow \\set{0}$ that is the identity on $\\set{0}$, sends $1$ to $0$, and is $t \\mapsto t^2 - t$ on $\\R$.\nIt commutes with the maps of the diagrams and is a weak equivalence at each object.\nThe pullback of the first is empty and that of the second is a point, while both homotopy pullbacks are contractible path spaces.\n\nFor a diagram $D$ of Kan complexes there is an explicit model, the Bousfield–Kan formula $\\operatorname{holim} D = \\int_{i \\in I} \\mathrm{Map}(N(I/i), D(i))$, where $N(I/i)$ is the nerve of the over category.\nHomotopy colimits are defined dually, as the left derived functor of the colimit, using the projective model structure.\n",
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