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  "body": "\n## Definition\n\nLet $p \\colon \\mathcal{E} \\to \\mathcal{B}$ be a [[functor|28NQ0KRHD3]].\nA morphism $\\varphi \\colon x \\to y$ of $\\mathcal{E}$ is *cartesian* if this holds.\nFor every morphism $\\psi \\colon z \\to y$ of $\\mathcal{E}$ and every morphism $g \\colon p(z) \\to p(x)$ of $\\mathcal{B}$ with $p(\\varphi) \\circ g = p(\\psi)$, there is a unique $\\chi \\colon z \\to x$ with $\\varphi \\circ \\chi = \\psi$ and $p(\\chi) = g$.\nThe functor $p$ is a *fibration*, and $\\mathcal{E}$ is *fibred* over $\\mathcal{B}$, if this holds.\nFor every object $y$ of $\\mathcal{E}$ and every morphism $f \\colon b \\to p(y)$ of $\\mathcal{B}$, there is a cartesian morphism $\\varphi \\colon x \\to y$ with $p(\\varphi) = f$.\n\n## Remarks\n\nThe cartesian morphism $\\varphi$ over $f$ is unique up to a unique isomorphism over $\\id_b$, and $x$ is often written $f^* y$.\nThe fibre over $b$ is the subcategory of objects over $b$ and morphisms over $\\id_b$.\nFibrations over $\\mathcal{B}$ correspond to pseudofunctors from $\\mathcal{B}^{\\mathrm{op}}$ to categories, by the Grothendieck construction.\n\nExample: let $\\mathcal{B}$ have [[pullbacks|Q05K04QJ71]], let $\\mathcal{E}$ be the arrow category whose objects are morphisms $e \\colon a \\to b$ of $\\mathcal{B}$, and let $p$ send $e$ to its codomain $b$.\nGiven $e \\colon a \\to b$ and $f \\colon b' \\to b$, the pullback square of $e$ along $f$ is a morphism of $\\mathcal{E}$ over $f$.\nThe universal property of the pullback says exactly that it is cartesian, so $p$ is a fibration.\n\nNon-example: let $\\mathcal{B}$ be the category with objects $0, 1$ and one non-identity arrow $f \\colon 0 \\to 1$.\nLet $\\mathcal{E}$ have one object $y$ and only its identity, and let $p(y) = 1$.\nNo object of $\\mathcal{E}$ lies over $0$, so there is no morphism over $f$ ending at $y$, and $p$ is not a fibration.\n",
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