nLab
contractible type

Context

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(-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
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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) 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

semantics

Contractible types

Idea

In homotopy type theory, the notion of contractible type is an internalization of the notion of contractible space / (-2)-truncated object.

Contractible types are also called of h-level 00. They represent the notion true in homotopy type theory.

Definition

We work in intensional type theory with dependent sums, dependent products, and identity types,

Definition

For XX a type, let

isContr(X) x:X y:X(y=x) isContr(X) \coloneqq \sum_{x\colon X} \prod_{y\colon X} (y=x)

be the dependent sum in one variable x:Xx : X over the dependent product on the other variable y:Xy \colon X of the x,yx,y-dependent identity type (x=y)(x = y).

We say that XX is a contractible type if isContr(X)isContr(X) is an inhabited type.

Remark

In propositions as types language, this can be pronounced as “there exists a point x:Xx\colon X such that every other point y:Xy\colon X is equal to xx.”

Under the homotopy-theoretic interpretation, it should be thought of as the type of contractions of XX — since the dependent product describes continuous functions, the paths from yy to xx depend continuously on yy and thus exhibit a contraction of XX to xx.

Proposition

A provably equivalent definition is the product type of XX with the isProp-type of XX:

isContr(X)X×isProp(X). isContr(X) \coloneqq X \;\times\; isProp(X) \,.

(Here of course we have to use a definition of isProp which doesn’t refer to isContrisContr).

Remark

This now says that XX is contractible iff XX is inhabited and an h-proposition.

Properties

Proposition

For any type AA, the type isContr(A)isContr(A) is an h-proposition. In particular, we can show isContr(A)isContr(isContr(A))isContr(A) \to isContr(isContr(A)): if a type is contractible, then its space of contractions is also contractible.

Proposition

A type is contractible if and only if it is equivalent to the unit type.

Categorical semantics

We discuss the categorical semantics of contractible types.

Let 𝒞\mathcal{C} be a locally cartesian closed category with sufficient structure to intepret all the above type theory. This means that CC has a weak factorization system with stable path objects, and that acyclic cofibrations are preserved by pullback along fibrations between fibrant objects. (We ignore questions of coherence, which are not important for this discussion.)

In this categorical semantics, the interpretation of a type A:Type\vdash A : Type is a fibrant object [A:Type][\vdash A : Type], which for short we will just write AA. The interpretation of the identity type x,y:A(x=y):Typex,y : A \vdash (x = y) : Type is as the path space object A IA×AA^I \to A \times A. The interpretation of isContr(A)isContr(A) is the object obtained by taking the dependent product of the path space object along one projection p 2:A×AAp_2 : A\times A\to A and then forgetting the remaining morphism to AA.

[isContr(A)]= p 2A I A *. [isContr(A)] \;\; = \;\; \array{ \prod_{p_2} A^I \\ \downarrow \\ A \\ \downarrow \\ * } \,.

The interpretation [a^:isContr(A)][\hat a : isContr(A)] of a term of isContr(A)isContr(A) is precisely a morphism a^:* p 2A I\hat a : * \to \prod_{p_2} A^I.

Proposition

Let 𝒞\mathcal{C} be a type-theoretic model category. Write [isContr(A)][isContr(A)] for the interpretation of isContr(A)isContr(A) in 𝒞\mathcal{C}. Then:

Global points *[isContr(A)]* \to [isContr(A)] in 𝒞\mathcal{C} are in bijection with contraction right homotopies of the object AA, hence to diagrams in 𝒞\mathcal{C} of the form

A η A I (id,const a) A×A, \array{ A &\stackrel{\eta}{\to}& A^I \\ & {}_{\mathllap{(id, const_a)}}\searrow & \downarrow \\ && A \times A } \,,

where const aconst_a is a morphism of the form A*aAA \to * \stackrel{a}{\to} A and where A IA^I is the path space object of AA in 𝒞\mathcal{C}.

Proof

Given a global point a^:* p 2A I\hat a : * \to \prod_{p_2} A^I, write a:*Aa : * \to A for the corresponding composite

* a^ p 2A I a A. \array{ * &\stackrel{\hat a}{\to} & \prod_{p_2} A^I \\ &{}_{\mathllap{a}}\searrow & \downarrow \\ && A } \,.

in 𝒞\mathcal{C}. This is an element in the hom set 𝒞 /A(a, p 2A I)\mathcal{C}_{/A}(a, \prod_{p_2} A^I) of the slice category over AA. By the (base change \dashv dependent product)-adjunction this is equivalently an element in 𝒞 /A×A(p 2 *a,A I)\mathcal{C}_{/A \times A}( p_2^* a, A^I ).

Notice that the pullback p 2 *ap_2^* a is the left morphism in

A * (id,const a) a A×A p 2 A. \array{ A &\to& * \\ {}^{\mathllap{(id,const_a)}}\downarrow && \downarrow^{\mathrlap{a}} \\ A \times A &\stackrel{p_2}{\to}& A } \,.

Therefore a morphism p 2 *aA Ip_2^* a \to A^I in 𝒞 /A×A\mathcal{C}_{/A \times A} is equivalently in 𝒞\mathcal{C} a diagram of the form

A η A I (id,const a) A×A. \array{ A &&\stackrel{\eta}{\to}&& A^I \\ & {}_{\mathllap{(id,const_a)}}\searrow && \swarrow \\ && A \times A } \,.

This is by definition a contraction right homotopy of AA.

Remark

Thus if isContr(A)isContr(A), then A1A\to 1 is a (right) homotopy equivalence, and hence (since AA is fibrant) an acyclic fibration.

Conversely, if 𝒞\mathcal{C} is a model category, AA and 11 are cofibrant, and A1A\to 1 is an acyclic fibration, then A1A\to 1 is a right homotopy equivalence, and hence isContr(A)isContr(A) has a global element. Thus, in most cases, the existence of a global element of isContr(A)isContr(A) (which is unique up to homotopy, since isContr(A)isContr(A) is an h-proposition) is equivalent to A1A\to 1 being an acyclic fibration.

More generally, we may apply this locally. Suppose that ABA\to B is a fibration, which we can regard as a dependent type

x:BA(x):Type.x\colon B \vdash A(x)\colon Type.

Then we have a dependent type

x:BisContr(A(x)):Typex\colon B \vdash isContr(A(x))\colon Type

represented by a fibration isContr(A)BisContr(A)\to B. By applying the above argument in the slice category 𝒞/B\mathcal{C}/B, we see that (if 𝒞\mathcal{C} is a model category, and AA and BB are cofibrant) isContr(A)BisContr(A)\to B has a section exactly when ABA\to B is an acyclic fibration.

We can also construct the type

x:BisContr(A(x))\prod_{x\colon B} isContr(A(x))

in global context, which has a global element precisely when isContr(A)BisContr(A)\to B has a section. Thus, a global element of this type is also equivalent to ABA\to B being an acyclic fibration.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoidh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-\infty-groupoid

References

Coq-code for contractible types is at

Revised on September 10, 2012 20:18:21 by Urs Schreiber (131.174.188.17)