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\begin{document}

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\section*{contractible type}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{contractible_types}{}\section*{{Contractible types}}\label{contractible_types}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{CategoricalSemantics}{Categorical semantics}\dotfill \pageref*{CategoricalSemantics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[homotopy type theory]], the notion of \textbf{contractible [[type]]} is an internalization of the notion of [[contractible space]] / [[(-2)-truncated]] object.

Contractible types are also called of \textbf{[[h-level]] $0$}. They represent the notion \emph{[[true]]} in homotopy type theory.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[type]], let

\begin{displaymath}
isContr(X) \coloneqq \sum_{x\colon X} \prod_{y\colon X} (y=x)
\end{displaymath}
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 type|dependent]] [[identity type]] $(x = y)$.

We say that $X$ is a \textbf{contractible type} if $isContr(X)$ is an [[inhabited type]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
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 theory|homotopy-theoretic]] interpretation, it should be thought of as the type of \emph{contractions} of $X$ --- since the dependent product describes \emph{continuous} functions, the paths from $y$ to $x$ depend continuously on $y$ and thus exhibit a contraction of $X$ to $x$.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
A provably [[equivalence in homotopy type theory|equivalent]] definition is the [[product type]] of $X$ with the [[isProp]]-type of $X$:

\begin{displaymath}
isContr(X) \coloneqq X \;\times\; isProp(X) 
  \,.
\end{displaymath}
\end{prop}
(Here of course we have to use a definition of [[isProp]] which doesn't refer to $isContr$).

\begin{remark}
\label{}\hypertarget{}{}
This now says that $X$ is contractible iff $X$ is [[inhabited type|inhabited]] and an [[h-proposition]].

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
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.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
A type is contractible if and only if it is [[equivalence in homotopy type theory|equivalent]] to the [[unit type]].

\end{prop}
\hypertarget{CategoricalSemantics}{}\subsection*{{Categorical semantics}}\label{CategoricalSemantics}

We discuss the [[categorical semantics]] of contractible types.

Let $\mathcal{C}$ be a [[locally cartesian closed category]] with [[categorical semantics of homotopy type theory|sufficient structure]] to intepret all the above type theory. This means that $\mathcal{C}$ 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 $A$.

\begin{displaymath}
[isContr(A)]
  \;\;
  =
  \;\;
  \itexarray{
    \prod_{p_2} A^I
    \\
    \downarrow
    \\
    A
    \\
    \downarrow
    \\
    *
  }
  \,.
\end{displaymath}
The interpretation $[\hat a : isContr(A)]$ of a [[term]] of $isContr(A)$ is precisely a morphism $\hat a : * \to \prod_{p_2} A^I$.

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[type-theoretic model category]]. Write $[isContr(A)]$ for the [[categorical semantics|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

\begin{displaymath}
\itexarray{
    A &\stackrel{\eta}{\to}& A^I
    \\
    & {}_{\mathllap{(id, const_a)}}\searrow & \downarrow
    \\
    && A \times A
  }
  \,,
\end{displaymath}
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 $\mathcal{C}$.

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

\begin{displaymath}
\itexarray{
     * &\stackrel{\hat a}{\to} & \prod_{p_2} A^I
     \\
     &{}_{\mathllap{a}}\searrow & \downarrow
     \\
     && A
  }
  \,.
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    A &\to& *
    \\
    {}^{\mathllap{(id,const_a)}}\downarrow && \downarrow^{\mathrlap{a}}
    \\
    A \times A &\stackrel{p_2}{\to}& A 
  }
  \,.
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    A &&\stackrel{\eta}{\to}&& A^I
    \\
    & {}_{\mathllap{(id,const_a)}}\searrow && \swarrow
    \\
    && A \times A
  }
  \,.
\end{displaymath}
This is by definition a [[contraction]] right [[homotopy]] of $A$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
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

\begin{displaymath}
x\colon B \vdash A(x)\colon Type.
\end{displaymath}
Then we have a dependent type

\begin{displaymath}
x\colon B \vdash isContr(A(x))\colon Type
\end{displaymath}
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

\begin{displaymath}
\prod_{x\colon B} isContr(A(x))
\end{displaymath}
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.

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include homotopy n-types - table]]

\begin{itemize}%
\item [[isEquiv]]


\item \textbf{isContr}


\item [[isProp]]


\item [[contractible space]], [[contractible chain complex]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

[[Coq]]-code for contractible types is at

\begin{itemize}%
\item \href{https://github.com/HoTT/HoTT/blob/master/Coq/Contractible.v}{HoTT repository}

\end{itemize}
[[!redirects contractible type]] [[!redirects contractible types]] [[!redirects h-level 0]] [[!redirects h-level 0 type]] [[!redirects (-2)-truncated type]] [[!redirects (-2)-truncated types]] [[!redirects type of contractions]] [[!redirects h-contractible type]] [[!redirects isContr]]



\end{document}