observational 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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type

falseinitial objectempty type

proposition(-1)-truncated objecth-proposition, mere proposition

proofgeneralized elementprogram

cut rulecomposition of classifying morphisms / pullback of display mapssubstitution

cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction

introduction rule for implicationunit for hom-tensor adjunctioneta conversion

logical conjunctionproductproduct type

disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)

implicationinternal homfunction type

negationinternal hom into initial objectfunction type into empty type

universal quantificationdependent productdependent product type

existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)

equivalencepath space objectidentity type

equivalence classquotientquotient type

inductioncolimitinductive type, W-type, M-type

higher inductionhigher colimithigher inductive type

completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set

setinternal 0-groupoidBishop set/setoid

universeobject classifiertype of types

modalityclosure operator, (idemponent) 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




What is called observational type theory (OTT) is a flavor of type theory in between extensional type theory and intensional type theory.

It may be regarded as a first-stage approximation to homotopy type theory (HoTT): the propositions of OTT correspond to the h-propositions of HoTT, and the types in OTT correspond to h-sets in HoTT. The notion of equality on OTT is based on setoids, which is a special case of higher internal groupoids. Since equality is defined type-by-type, OTT enjoys function extensionality, and a form of propositional extensionality at least for a specified universe of propositions (not necessarily including all h-propositions).

There are a few technical differences (e.g. proofs of propositions are definitionally equal in OTT, whereas proofs of hprops are only propositionally equal in HoTT) but on the whole HoTT looks a lot like a higher-dimensional version of OTT.


Observational type theory was introduced in

A blog post about an Agda implementation, including propositional extensionality (which is not mentioned in the above paper) is at

The above comparison between OTT and HoTT is taken from

Last revised on July 29, 2017 at 19:03:34. See the history of this page for a list of all contributions to it.