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  "body": "\n## Summary\n\nHomotopy type theory reads [[intuitionistic type theory|ESVYD9CZ8C]] in terms of homotopy theory: types are spaces, terms are points, and terms of identity types are paths.\nIt adds Voevodsky's univalence axiom, which says that equivalent types are equal, and higher inductive types such as the circle.\nThese additions give a new foundation of mathematics, called univalent foundations.\nThe book \"Homotopy Type Theory: Univalent Foundations of Mathematics\", written in 2013 at the Institute for Advanced Study, set out the theory.\n\n## Explanations\n\n### L1 human\n\nIn ordinary [[type theory|Q48KGV33AH]], a proof that $a = b$ is just a certificate.\nIn homotopy type theory, it is a path from $a$ to $b$ in a space, and two such paths need not be the same.\nA proof that two proofs of $a = b$ are equal is then a deformation of one path into the other.\nThe univalence axiom says that two types with a one-to-one correspondence between them are equal, so isomorphic structures can always be swapped.\nThis matches how mathematicians already treat isomorphic objects.\n",
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