nLab
type

This entry is about the notion in type theory. For the unrelated notion of the same name in model theory see at type (in model theory).


Context

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

equivalence classquotientquotient type

inductioncolimitinductive type, W-type, M-type

higher inductionhigher colimithigher inductive type

completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set

setinternal 0-groupoidBishop set/setoid

universeobject classifiertype of types

modalityclosure operator, (idemponent) 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

</table>

homotopy levels

semantics

Contents

Idea

In modern logic, we understand that every variable should have a type, or domain of discourse or be of some sort. For instance we say that if a variable nn is constrained to be an integer then “nn is of integer type” or “of type \mathbb{Z}”. The usual formal expression from set theory for this – nn \in \mathbb{Z} – is then often written n:n \colon \mathbb{Z}

We speak of typed logic if this typing of variables is enforced by the metalanguage. In formulations of a theory the types are often called sorts. More generally, type theory formalizes reasoning with such typed variables. See there for more

(Untyped logic may be seen as simply a special case, in which there is only a single unique type. Thus, untyped logic has one type, not no type.)

Definition

Reasoning with types is formalized in natural deduction (which in turn is formalized in a logical framework such as Elf).

Behaviour of types is specified by a 4-step set of rules

  1. type formation

  2. term introduction

  3. term elimination

  4. computation rules

Properties

Deep relations between type theory, category theory and computer science relate types to other notions, such as objects in a category. See at computational trinitarianism for more on this.

type, type theory

dependent type, dependent type theory, Martin-Löf dependent type theory

homotopy type, homotopy type theory

Last revised on April 27, 2017 at 09:01:46. See the history of this page for a list of all contributions to it.