Math 2.0

Mathematical logic and foundations Concept

Univalence axiom

Builds on Intuitionistic type theoryHomotopy type theoryHomotopy equivalence

Summary

The univalence axiom is an axiom of Martin-Löf type theory formulated by Vladimir Voevodsky. It says that equality of types in a universe is the same as equivalence of types. More precisely, the canonical map from identifications A=BA = B to equivalences A≃BA \simeq B is itself an equivalence. It is the central axiom of homotopy type theory and of univalent foundations. It makes every construction respect equivalence, so that isomorphic structures can be used interchangeably, and it implies function extensionality.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionHomotopy equivalence, concept, no evidence levelHomotopy equivalenceconceptIntuitionistic 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 theoryconceptUnivalence axiom, concept, no evidence levelUnivalence axiomconcept
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. Homotopy equivalence concept
  3. Intuitionistic logic concept
  4. Lambda calculus concept
  5. Proposition concept
  6. Set concept
  7. Set definition
  8. Conjunction definition
  9. Disjunction definition
  10. Implication definition
  11. Negation definition
  12. Biconditional definition
  13. Truth table concept
  14. Modus ponens concept
  15. Mathematical proof concept
  16. Propositional logic concept
  17. Tautology definition
  18. Logical equivalence definition
  19. Double negation law statement
  20. Proof by contradiction concept
  21. Russell's paradox statement
  22. Type theory concept
  23. Curry–Howard correspondence concept
  24. Intuitionistic type theory concept
  25. Homotopy type theory concept
  26. Univalence axiom concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Univalent_foundations
About this page
TagA53TSXZTW5
URIm2://A53TSXZTW5
Kindconcept
WritingW0 stub
MSC03B38, 55U35, 03F55
UsesIntuitionistic type theory, Homotopy type theory, Homotopy equivalence
Changed3 November 2026

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