Math 2.0

Mathematical logic and foundations Constructive mathematics Concept

Homotopy type theory

Builds on Intuitionistic type theoryType theory

Summary

Homotopy type theory reads intuitionistic type theory in terms of homotopy theory: types are spaces, terms are points, and terms of identity types are paths. It adds Voevodsky's univalence axiom, which says that equivalent types are equal, and higher inductive types such as the circle. These additions give a new foundation of mathematics, called univalent foundations. The book "Homotopy Type Theory: Univalent Foundations of Mathematics", written in 2013 at the Institute for Advanced Study, set out the theory.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionIntuitionistic logic, concept, no evidence levelIntuitionistic logicconceptLambda calculus, concept, no evidence levelLambda calculusconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptRussell's paradox, statement, no evidence levelRussell's paradoxstatementType theory, concept, no evidence levelType theoryconceptCurry–Howard correspondence, concept, no evidence levelCurry–Howard correspo…conceptIntuitionistic type theory, concept, no evidence levelIntuitionistic type t…conceptHomotopy type theory, concept, no evidence levelHomotopy type theoryconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Intuitionistic logic concept
  3. Lambda calculus concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Negation definition
  11. Biconditional definition
  12. Truth table concept
  13. Modus ponens concept
  14. Mathematical proof concept
  15. Propositional logic concept
  16. Tautology definition
  17. Logical equivalence definition
  18. Double negation law statement
  19. Proof by contradiction concept
  20. Russell's paradox statement
  21. Type theory concept
  22. Curry–Howard correspondence concept
  23. Intuitionistic type theory concept
  24. Homotopy type theory concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Homotopy_type_theory
About this page
TagEPAAKD80H9
URIm2://EPAAKD80H9
Kindconcept
WritingW0 stub
MSC03B38, 55U35, 03F55
UsesIntuitionistic type theory, Type theory
Same as
Changed2 November 2026

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