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Mathematical logic and foundations Definition

Univalence axiom

Builds on Intuitionistic type theoryDependent typeHomotopy type theory

Definition

Work in Martin-Löf type theory with dependent types Π\Pi and Σ\Sigma, identity types x=Ayx =_A y and a universe 𝒰\mathcal{U}. For f,g:Πx:AB(x)f, g : \Pi_{x : A} B(x), write f∼gf \sim g for the type Πx:A(f x=g x)\Pi_{x : A} (f\, x = g\, x). A function f:A→Bf : A \to B is an equivalence if the type isequiv(f)=(Σg:B→A g∘f∼idA)×(Σh:B→A f∘h∼idB)\mathrm{isequiv}(f) = \Big(\Sigma_{g : B \to A}\, g \circ f \sim \mathrm{id}_A\Big) \times \Big(\Sigma_{h : B \to A}\, f \circ h \sim \mathrm{id}_B\Big) has a term, and A≃BA \simeq B is the type Σf:A→Bisequiv(f)\Sigma_{f : A \to B} \mathrm{isequiv}(f). For A,B:𝒰A, B : \mathcal{U}, let idtoeqv:(A=𝒰B)→(A≃B)\mathrm{idtoeqv} : (A =_{\mathcal{U}} B) \to (A \simeq B) be the map defined by path induction that sends reflA\mathrm{refl}_A to the identity equivalence of AA. The univalence axiom for 𝒰\mathcal{U} states that for all A,B:𝒰A, B : \mathcal{U}, the map idtoeqv\mathrm{idtoeqv} is an equivalence.

Remarks

Example: Voevodsky's model of type theory in Kan simplicial sets, where types are Kan complexes and identity types are path spaces, satisfies the univalence axiom. So the axiom is consistent relative to the set theory in which this model is built.

Non-example: consider type theory with uniqueness of identity proofs, which says that any two terms of x=Ayx =_A y are equal. It does not satisfy univalence for a universe containing the booleans 𝟐\mathbf{2}. Univalence would turn the swap 𝟐≃𝟐\mathbf{2} \simeq \mathbf{2} and the identity 𝟐≃𝟐\mathbf{2} \simeq \mathbf{2} into two terms of 𝟐=𝒰𝟐\mathbf{2} =_{\mathcal{U}} \mathbf{2}. These terms are different because the swap is not the identity, but uniqueness of identity proofs says they are equal.

The inverse of idtoeqv\mathrm{idtoeqv} is often written ua\mathrm{ua}. Univalence implies function extensionality: two functions f,gf, g with f∼gf \sim g are equal. It is the basic new axiom of homotopy type theory.

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 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…conceptDependent type, concept, no evidence levelDependent typeconceptIntuitionistic type theory, concept, no evidence levelIntuitionistic type t…conceptHomotopy type theory, concept, no evidence levelHomotopy type theoryconceptUnivalence axiom, definition, no evidence levelUnivalence axiomdefinition
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. Dependent type concept
  24. Intuitionistic type theory concept
  25. Homotopy type theory concept
  26. Univalence axiom definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Univalent_foundations
About this page
TagCZH07FFC9D
URIm2://CZH07FFC9D
Kinddefinition
WritingW0 stub
MSC03B38, 55U35, 03F55
DefinesUnivalence axiom
UsesIntuitionistic type theory, Dependent type, Homotopy type theory
Changed3 November 2026

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