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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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




Originally, “simple type theory” was the name of the type theory introduced by Church (1940) (therefore often and more informatively: “Church’s type theory” or similar). This type theory allowed (only) function type-type formation (therefore often: “simply typed lambda-calculus”) based on two elementary types (a kind of type of natural numbers and a type of propositions).

In mild generalization, if one admits in addition product type-type formation then [e.g. Gunter (1992)] these are the type theories whose categorical semantics is in cartesian closed categories [Lambek & Scott (1986), Part I], see also at relation between category theory and type theory.

More generally, the term “simple type theory” has come to refer to any type theory whose type formations are not indexed, in that the judgment that a type AA is well-formed has no other inputs, understood in contrast to:


Original articles:

  • 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, §5 of: A Formulation of the Simple Theory of Types, The Journal of Symbolic Logic 5 2 (1940) 56-68 [doi:10.2307/2266170]

See also

Establishing the syntax/semantics relation between cartesian closed categories and simply-typed lambda calculi:

Lecture notes:

Textbook accounts:

See also:

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

  • Nathanael Arkor, Marcelo Fiore, Algebraic models of simple type theories: a polynomial approach, Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (2020) 88-101 [doi:10.1145/3373718.3394771, arXiv:2006.16949[

Last revised on July 13, 2023 at 13:35:21. See the history of this page for a list of all contributions to it.