nLab
type of propositions

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Context

Universes

universe

(in category theory/type theory/computer science)

of all homotopy types

of homotopy n-types

of 0-truncated types/h-sets

of (-1)-truncated types/h-propositions

resizing

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-level 1-type/h-prop
proofgeneralized elementprogram
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 monadmodal type theory, monad (in computer science)

homotopy levels

semantics

Contents

Idea

In type theory the type of propositions is what in the categorical semantics becomes the subobject classifier.

Its generalization from propositions to general types is the type of types in homotopy type theory.

References

Detailed discussion of the type of propositions in Coq is in

Revised on October 3, 2012 02:01:13 by Urs Schreiber (82.169.65.155)
Edit | Back in time (2 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: subobject classifier, essentially algebraic theory, negation, Grothendieck universe, universe in a topos, inaccessible cardinal, coherent logic, universe, (sub)object classifier in an (infinity,1)-topos, proposition, codomain fibration, display map, calculus of constructions, signature (in logic), Horn theory, type-theoretic definition of category, pure type system, modal logic, universe enlargement, lambda-calculus, type of propositions, type of types, Boolean hyperdoctrine, stack semantics, cohesive homotopy type theory, interval type, function extensionality, sub-(infinity,1)-category - internal formulation, positive type, univalence axiom, type-theoretic model category, Oberwolfach HoTT-Coq tutorial, coherent hyperdoctrine, negative type, beta-reduction, eta-conversion, universe polymorphism, Gallina specification language, function application, universe - contents, geometry of physics, typical ambiguity, type of n-types, type of h-sets, Prop