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  "body": "\n## Summary\n\nA morphism of varieties is a map between algebraic varieties that is given locally by polynomials, or by quotients of polynomials with nonvanishing denominators.\nMorphisms are the maps of the category of varieties, and an isomorphism is a morphism with an inverse morphism.\nBetween affine varieties, morphisms correspond exactly to homomorphisms of the coordinate rings in the opposite direction.\n\n## Explanations\n\n### L1 human\n\nA morphism is a map between solution sets of polynomial equations that is itself written with polynomials.\nThe map $t \\mapsto (t, t^2)$ from the line onto the parabola $y = x^2$ is a morphism, and so is its inverse $(x, y) \\mapsto x$.\nSo the line and the parabola are isomorphic varieties.\nThe map $t \\mapsto (t^2, t^3)$ onto the cusp $y^2 = x^3$ is a morphism and a bijection, but its inverse is not a morphism, so these two are not isomorphic.\n",
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