nLab polarity in 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




In type theory, a type may be classified by its “polarity”, either positive or negative, according to whether its term constructors or its term eliminators are regarded as primary, respectively. The idea is due to Jean-Marc Andreoli and Jean-Yves Girard.

Positive types are inductively defined by their term introduction rules and correspond, under categorical semantics, to objects defined as covariant representable functors (CSetC \to Set) such as colimits and left adjoints.

On the other hand, negative types are coinductively defined by their term elimination rules and correspond under categorical semantics to objects defined as contravariant representable functors (C opSetC^{op} \to Set) such as limits and right adjoints.

Note that polarity is not so much a property of the types themselves, but rather of their presentation by introduction and elimination rules: It may happen that a positive and a negative type are equivalent (this is the case, for example, for the product type in simply typed lambda calculus). In terms of categorical semantics this is simply the fact that a single object may exhibit multiple universal properties.

Generally, in effectful languages, the positive types are better behaved in call-by-value? evaluation strategies and negative types are better behaved in call-by-name? and other lazy evaluation? strategies. A category-theoretic explanation of this fact is that call-by-value? languages can be modeled by the Kleisli category of a monad TT on a category CC. Usually, the category CC will have both limits (negatives) and colimits (positives), however the canonical functor F:CC TF \colon C \to C_T is a left adjoint so it only generally preserves the colimits (positives). Dually, call-by-name languages can be modeled by the Kleisli category of a comonad WW and the canonical functor U:CC WU \colon C \to C_W is a right adjoint, and so preserves limits (negatives).


Last revised on December 17, 2022 at 15:25:56. See the history of this page for a list of all contributions to it.