nLab
observational type theory

Context

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

semantics

Contents

Idea

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. 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.

References

Observational type theory was introduced in

The above comparison between OTT and HoTT is taken from

Revised on November 30, 2012 02:05:44 by Toby Bartels (64.89.53.239)