contractible type

**natural deduction** metalanguage, practical foundations

**type theory** (dependent, intensional, observational type theory, homotopy type theory)

**computational trinitarianism** = **propositions as types** +**programs as proofs** +**relation type theory/category theory**

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 $0$**. They represent the notion *true* in homotopy type theory.

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

For $X$ a type, let

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

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

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

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

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

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

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

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

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

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

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

We discuss the categorical semantics of contractible types.

Let $\mathcal{C}$ be a locally cartesian closed category with sufficient structure to intepret all the above type theory. This means that $C$ 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 $\vdash A : Type$ is a fibrant object $[\vdash A : Type]$, which for short we will just write $A$. The interpretation of the identity type $x,y : A \vdash (x = y) : Type$ is as the path space object $A^I \to A \times A$. The interpretation of $isContr(A)$ is the object obtained by taking the dependent product of the path space object along one projection $p_2 : A\times A\to A$ and then forgetting the remaining morphism to $A$.

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

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

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

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

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

where $const_a$ is a morphism of the form $A \to * \stackrel{a}{\to} A$ and where $A^I$ is the path space object of $A$ in $\mathcal{C}$.

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

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

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

Notice that the pullback $p_2^* a$ is the left morphism in

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

Therefore a morphism $p_2^* a \to A^I$ in $\mathcal{C}_{/A \times A}$ is equivalently in $\mathcal{C}$ a diagram of the form

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

This is by definition a contraction right homotopy of $A$.

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

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

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

$x\colon B \vdash A(x)\colon Type.$

Then we have a dependent type

$x\colon B \vdash isContr(A(x))\colon Type$

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

We can also construct the type

$\prod_{x\colon B} isContr(A(x))$

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

homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|

h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |

h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |

h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |

h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |

h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |

h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |

h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |

h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |

Coq-code for contractible types is at

Revised on October 12, 2017 14:49:01
by Urs Schreiber
(77.56.177.247)