nLab function application



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




A function ff is defined by its association to each input value xx (belonging to some allowable domain of values) of an output value, usually denoted f(x)f(x) or fxf x. The process of passing from ff and xx to f(x)f(x) is called function application, and one speaks of applying ff to xx to produce f(x)f(x).

The determination of the allowable domain for xx, given ff, depends a bit on foundational choices. In type theory and structural set theory, all functions have a type (a function type, naturally) which specifies their domain and codomain. In material set theory, a function is sometimes defined to be simply a particular sort of set of ordered pairs, with its domain specified implicitly as the set of elements occurring as first components of some such pair. (However, even in material set theory it is sometimes important for a function to come with a specified domain and/or codomain, in which case it can be defined to be an ordered triple.)

Syntax versus semantics

In formalized logic and type theory, ff, xx, and f(x)f(x) are terms (or more precisely, metavariables standing for terms), and the process of function application is a rule of term formation. This is something which belongs to the realm of syntax. On propositions ((-1)-truncated types) this is the modus ponens deduction rule.

Under a denotational semantics, each of these terms denotes a particular object, and we also refer to the object denoted by f(x)f(x) as the result of applying the object denoted by ff to the object denoted by xx. For instance, in a material set-theoretic semantics, ff would denote a set of ordered pairs such that for any aa, there is at most one bb such that (a,b)f(a,b)\in f, and xx would denote some aa such that there does exist such a bb, and f(x)f(x) would denote that uniquely specified bb. The distinction between the terms ff, xx, and f(x)f(x) and what they denote is usually (and harmlessly) blurred in ordinary mathematical practice, but when studying logic and type theory formally it becomes important.

Under an operational semantics, by contrast, the “meaning” of the term f(x)f(x) lies in how it is “evaluated”. Usually this proceeds by beta-reduction and related rules. For instance, if ff is the term λx.x*x\lambda x. x*x and xx is the term s(s(0))s(s(0)) (the numeral two), then f(x)f(x) is (λx.x*x)(s(s(0)))(\lambda x.x*x)(s(s(0))) which beta-reduces to s(s(0))*s(s(0))s(s(0))*s(s(0)). The definition of ** can then be invoked to cause futher beta-reductions resulting in s(s(s(s(0))))s(s(s(s(0)))) (the numeral four).

Last revised on January 26, 2014 at 07:12:56. See the history of this page for a list of all contributions to it.