nLab line type


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




Similar to how the interval type is the abstract homotopy type for the unit interval and the circle type is the abstract homotopy type for the unit circle, the line type is the abstract homotopy type for the real line.


The line type A 1A^1 is the higher inductive type generated by

  • an element 0:A 10:A^1

  • an equivalence of types s:A 1A 1s:A^1 \simeq A^1

  • an identification p:0= A 1s(0)p:0 =_{A^1} s(0)

Equivalently, it is the higher inductive type generated by

  • an element 0:A 10:A^1

  • an equivalence of types s:A 1A 1s:A^1 \simeq A^1

  • for each x:A 1x:A^1, an identification p(x):x= A 1s(x)p(x):x =_{A^1} s(x)

Equivalently, it is the higher inductive type generated by

  • a function c:A 1c:\mathbb{Z} \to A^1

  • for each z:z:\mathbb{Z}, an identification p(z):c(z)= A 1c(s(z))p(z):c(z) =_{A^1} c(s(z))

where \mathbb{Z} is the integers type.


The line type is contractible; see section 8.1.5 of UFP13.

Relation to the integers

The line type is equivalent to the propositional truncation of the integers type.

Relation to the circle type

The circle type S 1S^1 is the coequalizer type of the pair of functions on the line type

A 1sid A 1A 1S 1 A^1\underoverset{\quad s \quad}{\mathrm{id}_{A^1}}{\rightrightarrows}A^1 \to S^1

This is the analogue in synthetic homotopy theory of the fact that the unit circle is the coequalizer of the pair of functions on the real line in classical homotopy theory:

()+1id 𝕊 1 \mathbb{R} \underoverset{\quad (-) + 1 \quad}{\mathrm{id}_\mathbb{R}}{\rightrightarrows}\mathbb{R} \to \mathbb{S}^1

 Relation to the real numbers

In real-cohesive homotopy type theory, the shape of the real numbers is equivalent to the line type ʃ()A 1\esh(\mathbb{R}) \simeq A^1.


The line type is defined as “homotopical reals” in section 8.1.5 of

Last revised on November 11, 2023 at 13:17:26. See the history of this page for a list of all contributions to it.