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  "body": "\n## Definition\n\nStart from [[intuitionistic type theory|KCV8RKBATN]] with intensional identity types, dependent function and pair types, and a hierarchy of universes.\n*Homotopy type theory* is this theory extended by two principles.\n\n1. The *univalence axiom*: for each universe $U$ and all types $A, B : U$, the canonical map $\\mathrm{idtoeqv} \\colon (A =_U B) \\to (A \\simeq B)$ is an equivalence.\n2. *Higher inductive types*: types given by generators for points and also for paths. An example is the circle $S^1$, generated by a point $\\mathrm{base} : S^1$ and a path $\\mathrm{loop} : \\mathrm{base} =_{S^1} \\mathrm{base}$.\n\nHere $A \\simeq B$ is the type of equivalences, that is, of functions $f \\colon A \\to B$ that have a two-sided inverse up to identification.\nThe theory is read homotopically.\nA type is a space, a term $a : A$ is a point, and a term $p : a =_A b$ is a path from $a$ to $b$.\n\n## Remarks\n\nExample: in homotopy type theory one proves that the type of loops $\\mathrm{base} =_{S^1} \\mathrm{base}$ is equivalent to the integers $\\Z$.\nThis matches the classical fact that the fundamental group of the circle is $\\Z$.\nNon-example: intuitionistic type theory with extensional identity types, where any two terms of $a =_A b$ are equal, is not compatible with univalence.\nThe type $\\mathbf{2}$ of booleans has two self-equivalences, the identity and the swap.\nWith univalence they give two paths $\\mathbf{2} = \\mathbf{2}$, and extensional identity types make these paths equal.\nThen the identity and the swap would be equal, which is false.\nUnivalence implies function extensionality: two [[functions|0XHDTE5YMZ]] that agree at each point are equal.\n",
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