nLab synthetic homotopy 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, (∞,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


Homotopy type theory



Homotopy type theory provides a synthetic formalization of homotopy theory in its modern and most powerful incarnation in the guise of (∞,1)-toposes.

Just like plain dependent type theory is the internal logic of locally cartesian closed categories, so homotopy type theory is the internal logic of locally cartesian closed (∞,1)-categories, and homotopy type theory with univalent type universes is the internal logic of elementary (∞,1)-toposes. This means that homotopy type theory provides a “structural” foundation of the kind that William Lawvere had found in topos theory, but refined to homotopy theory in the refined guise of (∞,1)-topos theory.

Homotopy type theory natively knows about the formalization of simplicial homotopy theory and higher topos theory without the need to first formalize several textbooks worth of material starting from bare set theory. For instance, the formal proof of the Blakers-Massey theorem (FFLL) does not need to begin by first formalizing what a simplicial set is, what a Kan fibrancy condition is, what the infinite tower of homotopy groups is, what weak homotopy equivalences are, what homotopy pushouts are, how they reflect in long exact sequences of homotopy groups; because all this is native to homotopy type theory. Accordingly, the proof, on top of being a formal proof, is elegantly transparent and of actual practical interest. Moreover, since the formal HoTT proof generalizes the traditional statement to more general (∞,1)-toposes, it is actually a genuine new mathematical result of genuine interest in modern homotopy theory, to practicing mathematicians not concerned about foundations.

Cubical synthetic homotopy theory

Mörtberg & Pujet (2020) make the case for the use of cubical versions of HoTT in synthetic homotopy theory since univalence and HITs are natively supported there, rather than axiomatically added as in HoTT. The path algebra in HoTT is made complicated by the fact that many equalities do not hold definitionally, even in the proof of simple results such as that the torus is equivalent to the product of two circles. The proof of this result is trivial in cubical type theory.


The above text of the Idea section follows Schreiber 14.

The idea of regarding homotopy type theory as synthetic homotopy theory is essentially Awodey's proposal:


Further discussion in homotopy type theory:

Therefore, most of the articles listed at

count as synthetic homotopy theory.

The use of cubical type theory for synthetic homotopy theory is discussed in:

Last revised on November 22, 2023 at 20:06:20. See the history of this page for a list of all contributions to it.