nLab prefunction

Contents

Context

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms

Category theory

(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

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
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
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
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 setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
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

semantics

Contents

Idea

In certain foundations of mathematics, the “functions” between sets or types or presets do not preserve the equality in the set/type/preset. Many times, the functions that do not preserve equality are called “prefunctions”, and a function is then defined to be a prefunction that satisfies function extensionality.

Definition

In set theory

Suppose that XX and YY are sets, a prefunction from XX to YY is a mapping that does not preserve equality: it is not true that a=ba = b always implies f(a)=f(b)f(a) = f(b). A function is defined as a prefunction that preserves equality; such a prefunction is said to be extensional.

For example, consider the identity prefunction on the set of pairs of positive numbers, Z +×Z +Z^+ \times Z^+ and the set of positive fractions Q +Q^+. From Z +×Z +Z^+ \times Z^+ to Q +Q^+, this is a function, since a/b=c/da/b = c/d if (a,b)=(c,d)(a,b) = (c,d). But from Q +Q^+ to Z +×Z +Z^+ \times Z^+, it is not a function, since (for example) 2/4=3/62/4 = 3/6 but (2,4)(3,6)(2,4) \neq (3,6). A related example is the operation of taking the numerator of a (positive) fraction; from Q +Q^+ to Z +Z^+, we may view this as a prefunction but not as a function, although it is a function on Z +×Z +Z^+ \times Z^+.

Given sets XX and YY, the function set from XX to YY is a subset of this set of prefunctions between XX and YY. Composition of prefunctions is also possible, but likewise does not preserve equality.

In type theory

In many foundations based on type theory, such as in Martin-Löf type theory, all types come equipped with an identity type which behaves similarly as equality does in sets. These types, therefore, are not presets in the strict sense, in that the latter do not carry any equality at all. However, functions between such types usually do not satisfy function extensionality, so that in the set-theoretic sense they are still like prefunctions.

Categorical semantics

The difference between functions and prefunctions in sets is modeled in category theory (categorical semantics) as the difference between a concrete category and a category with a functor U:CSetU:C \to Set which is not faithful.

See also

Last revised on May 18, 2022 at 15:36:19. See the history of this page for a list of all contributions to it.