nLab simple 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/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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


A version of type theory. In practice any type theory is called a simple type theory if type formation is not indexed, that is the judgment that a type AA is well-formed has no other inputs. Contrast polymorphic type theory, where types depend on a context of type variables or dependent type theory, where types depend on a context of more general variables.


  • Bertrand Russell, Mathematical Logic as Based on the Theory of Types, American Journal of Mathematics, Vol. 30, No. 3 (Jul., 1908), pp. 222-262

  • Alonzo Church, A Formulation of the Simple Theory of Types, The Journal of Symbolic Logic Vol. 5, No. 2 (Jun., 1940), pp. 56-68 (JSTOR)

  • W. Farmer, The seven virtues of simple type theory, Journal of Applied Logic, Vol. 6, No. 3. (September 2008), pp. 267–286.

  • J. Roger Hindley, Basic Simple Type Theory, Cambridge University Press, 2008

Last revised on August 23, 2018 at 10:02:49. See the history of this page for a list of all contributions to it.