nLab circle 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




The circle type is an axiomatization of the homotopy type of the (shape of) the circle in the context of homotopy type theory.


As a higher inductive type, the circle is given by

Inductive Circle : Type
  | base : Circle
  | loop : Id Circle base base

This says that the type is inductively constructed from

  1. a term of circle type whose interpretation is as the base point of the circle,

  2. a term of the identity type of paths between these two terms, which interprets as the 1-cell of the circle

    baseloopbase base \stackrel{loop}{\to} base \,

    Hence a non-constant path from the base point to itself.

In natural deduction

Let Id A(a,b)\mathrm{Id}_A(a, b) denote the identification type between elements a:Aa:A and b:Ab:A of type AA, and let hId x:A.B(x)(a,b,p,y,z)\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) denote the heterogeneous identification type between elements y:B(a)y:B(a) and z:B(b)z:B(b) of type family x:AB(x)x:A \vdash B(x), given elements a:Aa:A and b:Ab:A and identification p:Id A(a,b)p:\mathrm{Id}_A(a, b). Let apd x:A.B(x)(f,a,b,p)\mathrm{apd}_{x:A.B(x)}(f, a, b, p) denote the dependent function application of the dependent function f: x:AB(x)f:\prod_{x:A} B(x) to the identification p:Id A(a,b)p:\mathrm{Id}_A(a, b)

In the natural deduction formulation of dependent type theory, the circle type is given by the following inference rules:

Formation rules for the circle type:

ΓctxΓS 1type\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash S^1 \; \mathrm{type}}

Introduction rules for the circle type:

ΓctxΓ*:S 1ΓctxΓ𝓁:Id S 1(*,*)\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash *:S^1} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathcal{l}:\mathrm{Id}_{S^1}(*,*)}

Elimination rules for the circle type:

Γ,x:S 1C(x)typeΓc *:C(*)Γc 𝓁:hId x:S 1.C(x)(*,*,𝓁,c *,c *)Γ,x:S 1ind S 1 x:S 1.C(x)(c *,c 𝓁)(x):C(x)\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_\mathcal{l}:\mathrm{hId}_{x:S^1.C(x)}(*, *, \mathcal{l}, c_*, c_*)}{\Gamma, x:S^1 \vdash \mathrm{ind}_{S^1}^{x:S^1.C(x)}(c_*, c_\mathcal{l})(x):C(x)}

Computation rules for the circle type:

Γ,x:S 1C(x)typeΓc *:C(*)Γc 𝓁:hId x:S 1.C(x)(*,*,𝓁,c *,c *)Γβ S 1 *(c *,c 𝓁):Id C(*)(ind S 1 x:S 1.C(x)(c *,c 𝓁)(*),c *)\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_\mathcal{l}:\mathrm{hId}_{x:S^1.C(x)}(*, *, \mathcal{l}, c_*, c_*)}{\Gamma \vdash \beta_{S^1}^{*}(c_*, c_\mathcal{l}):\mathrm{Id}_{C(*)}(\mathrm{ind}_{S^1}^{x:S^1.C(x)}(c_*, c_\mathcal{l})(*), c_*)}
Γ,x:S 1C(x)typeΓc *:C(*)Γc 𝓁:hId x:S 1.C(x)(*,*,𝓁,c *,c *)Γβ S 1 𝓁(c *,c 𝓁):Id hId x:S 1.C(x)(*,*,𝓁,c *,c *)(apd x:S 1.C(x)(ind S 1 x:S 1.C(x)(c *,c 𝓁)),c 𝓁,*,*,𝓁)\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_\mathcal{l}:\mathrm{hId}_{x:S^1.C(x)}(*, *, \mathcal{l}, c_*, c_*)}{\Gamma \vdash \beta_{S^1}^{\mathcal{l}}(c_*, c_\mathcal{l}):\mathrm{Id}_{\mathrm{hId}_{x:S^1.C(x)}(*, *, \mathcal{l}, c_*, c_*)}(\mathrm{apd}_{x:S^1.C(x)}(\mathrm{ind}_{S^1}^{x:S^1.C(x)}(c_*, c_\mathcal{l})), c_\mathcal{l}, *, *, \mathcal{l})}

Uniqueness rules for the circle type:

Γ,x:S 1C(x)typeΓc: x:S 1C(x)Γp:S 1Γl:Id S 1(p,p)Γη S 1(c,p,l):Id C(p)(c,ind S 1 x:S 1.C(x)(c(p),apd x:S 1.C(x)(c,p,p,l))\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:S^1} C(x) \quad \Gamma \vdash p:S^1 \quad \Gamma \vdash l:\mathrm{Id}_{S^1}(p, p)}{\Gamma\vdash \eta_{S^1}(c, p, l):\mathrm{Id}_{C(p)}(c,\mathrm{ind}_{S^1}^{x:S^1.C(x)}(c(p), \mathrm{apd}_{x:S^1.C(x)}(c, p, p, l))}

The elimination, computation, and uniqueness rules for the circle type state that the circle type satisfy the dependent universal property of the circle type. If the dependent type theory also has dependent sum types and product types, allowing one to define the uniqueness quantifier, the dependent universal property of the circle type could be simplified to the following rule:

Γ,x:S 1C(x)typeΓc *:C(*)Γc 𝓁:hId x:S 1.C(x)(*,*,𝓁,c *,c *)Γup S 1 x:S 1.C(x)(c *,c 𝓁):!c: x:S 1C(x).Id C(*)(c(*),c *)×Id hId x:S 1.C(x)(*,*,𝓁,c *,c *)(apd x:S 1.C(x)(c,c *,c *,c 𝓁),c 𝓁)\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_\mathcal{l}:\mathrm{hId}_{x:S^1.C(x)}(*, *, \mathcal{l}, c_*, c_*)}{\Gamma \vdash \mathrm{up}_{S^1}^{x:S^1.C(x)}(c_*, c_\mathcal{l}):\exists!c:\prod_{x:S^1} C(x).\mathrm{Id}_{C(*)}(c(*), c_*) \times \mathrm{Id}_{\mathrm{hId}_{x:S^1.C(x)}(*, *, \mathcal{l}, c_*, c_*)}(\mathrm{apd}_{x:S^1.C(x)}(c, c_*, c_*, c_\mathcal{l}), c_\mathcal{l})}

As a suspension

The circle type could also be defined as the suspension type Σ2\Sigma \mathbf{2} of the type of booleans 2\mathbf{2}.

As a coequalizer

The circle type could also be defined as the coequalizer type of any two endofunctions on the unit type

11S 1\mathbf{1} \rightrightarrows \mathbf{1} \to S^1

Using torsors

The circle can also be defined without HITs using only univalence, as the type of \mathbb{Z}-torsors. One can then prove that this type satisfies the same induction principle (propositionally). This is due to Dan Grayson.


Induction and recursion principles

Its induction principle says (e.g. UFP13, p. 177) that for

  • any P:S 1TypeP \colon S^1 \to Type

  • equipped with a point base:P(base)base' \colon P(base)

  • and a dependent path loop:base= loopbaseloop' \,\colon\, base' =_{loop} base',

there is f: (x:S 1)P(x)f:\prod_{(x:S^1)} P(x) such that:

f(base)=baseapd f(loop)=loopf(base)=base'\qquad apd_f(loop) = loop'

As a special case, its recursion principle says that given any type XX with a point x:Xx:X and a loop l:x=xl:x=x, there is f:S 1Xf:S^1 \to X with

f(base)=xap f(loop)=lf(base)=x\qquad ap_f(loop)=l

Extensionality principle of the circle type

The extensionality principle of the circle type says that there is an equivalence of types between the identification type Id S 1(*,*)\mathrm{Id}_S^1(*,*) and the type of integers \mathbb{Z}:

ext S 1:Id S 1(*,*)\mathrm{ext}_S^1:\mathrm{Id}_S^1(*,*) \simeq \mathbb{Z}

Equivalently, that the loop space type Ω(S 1,*)\Omega(S^1, *) is equivalent to the integers.

Thus, the extensionality principle of the circle type implies that the integers and thus the natural numbers are contractible types if axiom K or uniqueness of identity proofs hold for the entire type theory. If the extensionality principle of the natural numbers also hold in the type theory, then every type is contractible.

One could prove the extensionality principle of the circle type, given a univalent universe where the circle is small relative to the universe. The HoTT book provides a number of such proofs.

H-space structures on the circle type

The type of H-spaces on the circle type is a contractible type.


The formalization of the shape homotopy type ʃS 1Bʃ S^1 \simeq \mathbf{B}\mathbb{Z} of the circle as a higher inductive type in homotopy type theory, along with a proof that ΩʃS 1\Omega ʃS^1\simeq {\mathbb{Z}} (and hence π 1(ʃS 1)\pi_1(ʃS^1) \simeq \mathbb{Z}):

Formalization in proof assistants:

in Coq:

in Agda:

Exposition and review:

Alternative construction of the circle type as the type of \mathbb{Z}-torsors:

Alternative construction of the circle type as a coequalizer:

For the fact that the type of H-space structures on a circle type is contractible:

Last revised on August 2, 2023 at 12:44:13. See the history of this page for a list of all contributions to it.