nLab
Kripke-Joyal semantics

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

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-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

Kripke–Joyal semantics is a natural semantics in a topos.

The Mitchell–Bénabou language of a topos is a higher-order type theory in which one can write ordinary mathematics, which can then be interpreted in the topos using the Kripke–Joyal semantics.

References

Textbook introductions include

section VI.6 of

Lecture notes include

(page 58 and following)

Revised on November 10, 2011 03:07:30 by Toby Bartels (64.89.53.18)
Edit | Back in time (1 revision) | See changes | History | Views: Print | TeX | Source | Linked from: Timeline of category theory and related mathematics, function, logic, inhabited set, internal logic, context, homotopy n-type, essentially algebraic theory, homotopy type, negation, torsor, type theory, coherent logic, sequent calculus, vector bundle, computer science, type, preset, propositions as types, syntax - semantics duality, proposition, theory, dependent type, W-type, type theory - contents, first-order hyperdoctrine, tripos, identity type, display map, calculus of constructions, signature (in logic), Horn theory, algebraic model for modal logics, type-theoretic definition of category, pure type system, the logic S4(m), modal logic, normal modal logic, locally ringed topos, term, variable, cartesian logic, hyperdoctrine, categorical semantics, lambda-calculus, syntactic category, cartesian theory, axiom, homotopy type theory, type of propositions, theorem, directed homotopy type theory, unit type, type of types, axiom UIP, Boolean hyperdoctrine, real numbers object, contents of contents, syntax, product type, quotient type, stack semantics, cohesive homotopy type theory, interval type, homotopy level, function type, bracket type, dependent product type, Coq, empty type, inductive type, function extensionality, Agda, higher inductive type, existential quantifier, universal quantifier, sub-(infinity,1)-category - internal formulation, positive type, proofs as programs, proof, HoTT methods for homotopy theorists, univalence axiom, intensional type theory, type-theoretic model category, judgmental equality, dependent type theory, contractible type, extensional type theory, sum type, reflective sub-(infinity,1)-category - internal formulation, semantics, Oberwolfach HoTT-Coq tutorial, h-set, decidability, h-proposition, equivalence in homotopy type theory, canonical form, polymorphism, coherent hyperdoctrine, negative type, type formation, beta-reduction, eta-conversion, dependent sum type, substitution, fixed-point combinator, relation between type theory and category theory, M-type, categorical model of dependent types, sequent, first-order theory, Haskell, logical framework, Burali-Forti's paradox, Gallina specification language, judgment, Russell's paradox, context extension, function application, numeral, definition, program, elementary function arithmetic, Brouwer-Heyting-Kolmogorov interpretation, second-order arithmetic, computational trinitarianism, monad (in computer science), computation, programming language, natural deduction, metalanguage, theory of presheaf type, Practical Foundations for Programming Languages, term in context, contradiction, inconsistency, modal type, type of h-sets, Bishop set, antecedent, succedent, consequent, looping combinator, deductive system, h-groupoid, coinductive type, practical foundation of mathematics, modal type theory, differential homotopy type theory, axiom K, observational type theory, meaning explanation, natural numbers type