nLab Introduction to Homotopy Type Theory


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


This entry collects links related to the forthcoming book

  • Egbert Rijke:

    Introduction to Homotopy Type Theory

    Cambridge Studies in Advanced Mathematics,

    Cambridge University Press

    arXiv:2212.11082 (359 pages)

which introduces homotopy type theory in general and in particular Martin-Löf's dependent type theory, the Univalent Foundations for Mathematics and synthetic homotopy theory.

The book is based on a course taught by the author at Carnegie Mellon University in the spring semester of 2018:

The original writeup of these course notes

contains considerably more material than retained in the above arXiv version.

Other versions:


Chapter I introduces the reader to Martin-Löf's dependent type theory. The fundamental concepts of type theory are explained without immediately jumping into the homotopy interpretation of type theory.

1.\;\;\;1. Dependent Type Theory

2.\;\;\;2. Dependent function types

3.\;\;\;3. The natural numbers

4.\;\;\;4. More inductive types

5.\;\;\;5. Identity types

6.\;\;\;6. Universes

7.\;\;\;7. Modular arithmetic via the Curry-Howard isomorphism

8.\;\;\;8. Decidability in elementary number theory

Chapter II is an exposition of the Univalent Foundations for Mathematics. This chapter gradually extends dependent type theory with function extensionality, propositional truncation, the univalence axiom, and the type theoretic replacement axiom.

9.\;\;\;9. Equivalences

10.\;\;\;10. Contractible types

11.\;\;\;11. The fundamental theorem of identity types

12.\;\;\;12. Propositions, sets, and the higher truncation levels

13.\;\;\;13. Function extensionality

14.\;\;\;14. Propositional truncations

15.\;\;\;15. Image factorizations

16.\;\;\;16. Finite types

17.\;\;\;17. The univalence axiom

18.\;\;\;18. Set quotients

19.\;\;\;19. Groups in univalent mathematics

20.\;\;\;20. General inductive types

Chapter III studies the circle as a higher inductive type.

21.\;\;\;21. The circle

22.\;\;\;22. The universal cover of the circle

An older version of the book ended:

22.\;\;\;22. Homotopy pullbacks

23.\;\;\;23. Homotopy pushouts

24.\;\;\;24. Cubical diagrams

25.\;\;\;25. Universality and descent for pushouts

26.\;\;\;26. Sequential colimits

27.\;\;\;27. Homotopy groups of types

28.\;\;\;28. The classifying type of a group

29.\;\;\;29. The Hopf fibration

30.\;\;\;30. The real projective spaces

31.\;\;\;31. Truncations

32.\;\;\;32. Connected types and maps

33.\;\;\;33. The Blakers-Massey theorem

34.\;\;\;34. Higher group theory


category: reference

Last revised on January 3, 2023 at 22:58:15. See the history of this page for a list of all contributions to it.