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  "body": "\n## Summary\n\nA holomorphic map between [[Riemann surfaces|5K0N8QMV6Q]] is a continuous map that is holomorphic when written in local coordinates on both sides.\nThese maps are the morphisms of the category of Riemann surfaces; the bijective ones are the biholomorphisms, which identify two Riemann surfaces.\nMeromorphic functions on a Riemann surface are the same as holomorphic maps to the [[Riemann sphere|0CDMXGPSES]] that are not constantly $\\infty$.\n\n## Explanations\n\n### L1 human\n\nOn a Riemann surface you cannot differentiate directly, but you can always look through a chart, which is a piece of the complex plane.\nA map between two surfaces is holomorphic if every such view of it is an ordinary holomorphic function.\nFor instance $z \\mapsto z^2$ on the Riemann sphere, with $\\infty \\mapsto \\infty$, is holomorphic: near $\\infty$ in the coordinate $w = 1/z$ it reads $w \\mapsto w^2$.\nNear any point where it is not constant, a holomorphic map looks like $z \\mapsto z^k$ for some $k \\ge 1$ in suitable coordinates.\n",
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