nLab center of contraction


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 a dependent type theory with identity types, a term a:Aa:A is a center of contraction or a centre of contraction if in the context of a variable b:Ab:A there is an identification p(b):a= Abp(b):a =_A b. If the type theory also has dependent product types, the above is equivalent to having a dependent function

p: b:Aa= Abp:\prod_{b:A} a =_A b

called a contraction of AA at aa. Thus, contractons of AA at aa are witnesses that a:Aa:A is a center of contraction.

We then define the type of contractions of AA at aa as

Contr A(a) b:Aa= Ab\mathrm{Contr}_A(a) \coloneqq \prod_{b:A} a =_A b

Rules for contraction types

If the dependent type theory does not have dependent product types, contraction types could still be defined by adding the formation, introduction, elimination, computation, and uniqueness types for contraction types

Formation rules for contraction types:

ΓAtypeΓ,a:A,x:Aa= AxtypeΓ,a:AContr A(a)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, a:A, x:A \vdash a =_A x \; \mathrm{type}}{\Gamma, a:A \vdash \mathrm{Contr}_A(a) \; \mathrm{type}}

Introduction rules for contraction types:

ΓAtypeΓ,a:A,x:Aa= AxΓ,a:Aλ(x).p(x):Contr A(a)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, a:A, x:A \vdash a =_A x}{\Gamma, a:A \vdash \lambda(x).p(x):\mathrm{Contr}_A(a)}

Elimination rules for contraction types:

Γ,a:Ap:Contr A(a)Γb:AΓp(b):a= Ab\frac{\Gamma, a:A \vdash p:\mathrm{Contr}_A(a) \quad \Gamma \vdash b:A}{\Gamma \vdash p(b):a =_A b}

Computation rules for contraction types:

Γ,a:A,x:Ap(x):a= AxΓb:AΓβ Contr:(λ(x).p(x))(b)= a= Abp(b)\frac{\Gamma, a:A, x:A \vdash p(x):a =_A x \quad \Gamma \vdash b:A}{\Gamma \vdash \beta_\mathrm{Contr}:(\lambda(x).p(x))(b) =_{a =_A b} p(b)}

Uniqueness rules for contraction types:

Γ,a:Ap:Contr A(a)Γη Contr:p= Contr A(a)λ(x).p(x)\frac{\Gamma, a:A \vdash p:\mathrm{Contr}_A(a)}{\Gamma \vdash \eta_\mathrm{Contr}:p =_{\mathrm{Contr}_A(a)} \lambda(x).p(x)}


A type AA is a contractible type if there exists a center of contraction

isContr(A)= a:AContr A(a)isContr(A) = \sum_{a:A} \mathrm{Contr}_A(a)

and a type AA is an h-proposition if every element in AA is a center of contraction

isProp(A)= a:AContr A(a)isProp(A) = \prod_{a:A} \mathrm{Contr}_A(a)

The axiom K on a type states that for every element a:Aa:A, reflexivity refl A(a)\mathrm{refl}_A(a) is the center of contraction of the loop space type of aa.

See also


Last revised on January 31, 2024 at 16:26:02. See the history of this page for a list of all contributions to it.