nLab layered 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 two-level type theory, there are also two different kinds of judgments, one for the types representing the metatheory, and one for the types representing the object theory. In typed predicate logic, there are two different kinds of judgments, one for types and one for propositions. In type theory with shapes, there are three different kinds of judgments, one for cubes, one for topes, and one for types, where the subtheory of cubes and topes together form a predicate logic over a type theory with finite product types. Examples of type theory with shapes include certain presentations of simplicial type theory and cubical type theory. Independently, cubical type theory could be considered as a type theory over a typed predicate logic with only one type, where the single type in the predicate logic is called the “interval” and the propositions in the predicate logic layer are called “face formulas”.

Thus, one could think of the different judgments as forming different layers of the type theory, and these type theories as layered type theory.

Some layered type theories have split contexts, in which the contexts (and associated judgments) sit in a sequential hierarchy: for example, in two-level type theory, the types in the object theory can depend upon variables in the metatheory, but the types in the metatheory cannot depend upon variables in the object theory.

Layered type theories could be contrasted with theories like the usual presentations of Martin-Löf type theory and higher observational type theory, which only have one layer and thus are considered to be unlayered type theory.



The notion of layer of a type theory appears in

when defining type theory with shapes and simplicial type theory.

The term “layered type theory” also appears in

  • César Bardomiano Martínez, Limits and colimits of synthetic \infty-categories [[arXiv:2202.12386]]

Last revised on January 18, 2023 at 00:50:13. See the history of this page for a list of all contributions to it.