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  "body": "\n## Definition\n\nLet $i \\colon A \\to B$ and $p \\colon X \\to Y$ be morphisms in a [[category|A2DE66937D]] $\\mathcal{C}$.\nThe morphism $i$ has the *left lifting property* with respect to $p$ if this holds.\nFor all morphisms $u \\colon A \\to X$ and $v \\colon B \\to Y$ with $p \\circ u = v \\circ i$, there is a morphism $d \\colon B \\to X$ with $d \\circ i = u$ and $p \\circ d = v$.\nIn that case $p$ is said to have the *right lifting property* with respect to $i$.\nIf the morphism $d$ is always unique, $i$ and $p$ are *orthogonal*.\n\n## Remarks\n\nThe data $u, v$ form a [[commutative square|WV898P4X16]], and $d$ is a diagonal filler of the square.\nA common notation is $i \\pitchfork p$, or $i \\perp p$ for orthogonality.\nFix a class $M$ of morphisms.\nThe morphisms with the left lifting property with respect to all members of $M$ contain all isomorphisms and are closed under composition.\n\nExample: in the category of sets, let $i \\colon \\emptyset \\to \\set{*}$ and let $p \\colon X \\to Y$ be surjective.\nA square is then just a point $y = v(*)$ of $Y$, and a filler is a point $x$ of $X$ with $p(x) = y$.\nIt exists since $p$ is surjective.\n\nNon-example: with the same $i$, let $p \\colon \\emptyset \\to \\set{*}$.\nTake $u$ the empty map and $v = \\id$; a filler would be a map $\\set{*} \\to \\emptyset$, which does not exist.\n\nIn topology, a Hurewicz fibration is a map with the right lifting property with respect to all inclusions $Z \\times \\set{0} \\to Z \\times [0, 1]$; this is the [[homotopy lifting property|G4QDM01P6J]].\n",
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