nLab type formation



Deduction and Induction

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, truth value(-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/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
set of truth valuessubobject classifiertype of propositions
universeobject 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 formation rule is a natural deduction step roughly of the form

A:TypeB:TypeF(A,B,):Type \frac{ \vdash\; A \colon Type \;\;\; \vdash \;B \colon Type \;\;\; \cdots }{ \vdash \; F(A,B,\cdots) \colon Type }

which says that given types A,B,A, B, \cdots , there is a new type F(A,B,)F(A,B, \cdots).

For instance for product types it says that

A:TypeB:TypeA×B:Type. \frac{\vdash \; A \colon Type \;\;\; \vdash \; B \colon Type}{\vdash \; A \times B \colon Type} \,.

In natural deduction the type formation rule for a kind of type is the first in a quadruple of rules that come with every kind of type:

  1. type formation rule

  2. term introduction rule

  3. term elimination rule

  4. computation rule


Relation to category theory

Under the relation between type theory and category theory, the categorical semantics of type formation essentially corresponds to certain universal constructions in category theory.


Introductory textbook account:

Last revised on September 24, 2022 at 08:20:04. See the history of this page for a list of all contributions to it.