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\begin{document}

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\section*{equivalence in homotopy type theory}

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

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

[[!include type theory - contents]]

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - contents]]

\hypertarget{equivalences_in_type_theory}{}\section*{{Equivalences in type theory}}\label{equivalences_in_type_theory}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Semantics}{Semantics}\dotfill \pageref*{Semantics} \linebreak
\noindent\hyperlink{of_}{Of $isEquiv(-)$}\dotfill \pageref*{of_} \linebreak
\noindent\hyperlink{of__2}{Of $Equiv(-)$}\dotfill \pageref*{of__2} \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{equivalence} is an internalization of the notion of [[equivalence]] or [[homotopy equivalence]].

These are sometimes called \textbf{[[weak equivalences]]}, but there is nothing [[weak equivalence|weak]] about them (in particular, they always have homotopy inverses).

\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 $f\colon A\to B$ a [[term]] of [[function type]]; we define new [[dependent types]] as follows:

the [[homotopy fiber]]

\begin{displaymath}
f \colon A \to B, b\colon B \vdash hfiber(f,b) \coloneqq \sum_{a\colon A} (f(a) = b)
\end{displaymath}
and the [[proposition]] that the homotopy fiber is a (dependently) [[contractible type]]:

\begin{displaymath}
f \colon A \to B 
   \vdash 
   isEquiv(f) \coloneqq \prod_{b\colon B} isContr(hfiber(f,b))
  \,.
\end{displaymath}
We say $f$ is an \textbf{equivalence} if $isEquiv(f)$ is an [[inhabited type]].

\end{defn}
That is, a function is an equivalence if all of its [[homotopy fibers]] are [[contractible types]] (in a way which depends continuously on the base point).

\begin{defn}
\label{}\hypertarget{}{}
For $X, Y : Type$ two [[types]], the \textbf{type of equivalences} from $X$ to $Y$ is the [[dependent sum]]

\begin{displaymath}
Equiv(X,Y)
  =
  (X \stackrel{\simeq}{\to}Y)
  \coloneqq
  \sum_{f : (X \to Y)}
  isEquiv(f)
  \,.
\end{displaymath}
\end{defn}
Three variations of this definition are, informally:

\begin{itemize}%
\item $f\colon A\to B$ is an equivalence if there is a map $g\colon B\to A$ and homotopies $p\colon \prod_{a\colon A} (g(f(a)) = a)$ and $q\colon \prod_{b\colon B} (f(g(b)) = b)$ (a \textbf{[[homotopy equivalence]]})


\item $f\colon A\to B$ is an equivalence if there is the above data, together with a higher homotopy expressing one [[triangle identity]] for $f$ and $g$ (an \textbf{[[adjoint equivalence]]}).


\item $f\colon A\to B$ is an equivalence if there are maps $g,h\colon B\to A$ and homotopies $p\colon \prod_{a\colon A} (g(f(a)) = a)$ and $q\colon \prod_{b\colon B} (f(h(b)) = b)$ (sometimes called a \textbf{homotopy isomorphism}).



\end{itemize}
By formalizing these, we obtain types $homotopyEquiv(f)$, $isAdjointEquiv(f)$, and $isHIso(f)$. All four of these types are co-inhabited: we have a function from any one of them to any of the others. Moreover, at least if we assume [[function extensionality]], the types $isAdjointEquiv(f)$ and $isHIso(f)$ are themselves \emph{equivalent} to $isEquiv(f)$, and all three are [[h-propositions]].

This is not true for $homotopyEquiv(f)$, which is not in general an h-prop even with function extensionality. However, often the most convenient way to show that $f$ is an equivalence is by exhibiting a term in $homotopyEquiv(f)$ (although such a term could just as well be interpreted to lie in $isHIso(f)$ with $h\coloneqq g$).

\hypertarget{Semantics}{}\subsection*{{Semantics}}\label{Semantics}

We discuss the [[categorical semantics]] of equivalences in homotopy type theory.

Let $\mathcal{C}$ be a [[locally cartesian closed category]] which is a [[model category]], in which the (acyclic cofibration, fibration) [[weak factorization system]] has [[stable path objects]], and acyclic cofibrations are preserved by pullback along fibrations between fibrant objects. (We ignore questions of coherence, which are not important for this discussion.) For instance $\mathcal{C}$ could be a [[type-theoretic model category]].

\hypertarget{of_}{}\subsubsection*{{Of $isEquiv(-)$}}\label{of_}

\begin{prop}
\label{}\hypertarget{}{}
For $A, B$ two [[cofibrant object|cofibrant]]-[[fibrant objects]] in $\mathcal{C}$, a morphism $f\colon A\to B$ is a [[weak equivalence]] or equivalently a [[homotopy equivalence]] in $\mathcal{C}$ precisely when the interpretation of $isEquiv(f)$ has a [[generalized element|global point]] $* \to isEquiv(f)$.

\end{prop}
\begin{proof}
For $f\colon A\to B$, the [[categorical semantics]] of the [[dependent type]]

\begin{displaymath}
b\colon B 
   \;\vdash\; 
  hfiber(f,b)\colon Type
\end{displaymath}
is by the rules for the [[categorical semantics|interpretation]] of [[identity types]] and [[substitution]] the [[mapping path space]] construction $P f$, given by the [[pullback]]

\begin{displaymath}
\itexarray{
    [b : B \vdash  hfiber(f,b)] &\coloneqq & P f  &\to& A
    \\
    && \downarrow && \downarrow^{\mathrlap{f}}
    \\
    && B^I &\to& B
    \\
    && \downarrow
    \\
    && B
  }
\end{displaymath}
which, by the [[factorization lemma]], is one way to factor $f$ as an [[acyclic cofibration]] followed by a [[fibration]]

\begin{displaymath}
f : A \stackrel{\simeq}{\to} P f \to B
  \,.
\end{displaymath}
By definition and the semantics of [[contractible types]], therefore, if $A$ and $B$ are cofibrant, then $isEquiv(f)$ has a [[global element]]

\begin{displaymath}
* \to \prod_{b} isContr(hfiber(f,b))
\end{displaymath}
precisely when in this factorization, the fibration $P f \to B$ is an acyclic fibration. (See for instance (\href{http://www.sandiego.edu/~shulman/hottminicourse2012/03models.pdf#page=49}{Shulman, page 49}) for more details.)

But by the [[2-out-of-3 property]], this is equivalent to $f$ being a weak equivalence --- and hence a homotopy equivalence, since it is a map between fibrant-cofibrant objects.

\end{proof}
\begin{remark}
\label{InterpretationIfIsEquivAsDependentType}\hypertarget{InterpretationIfIsEquivAsDependentType}{}
In the above we fixed one function $f : A \to X$. But the type $isEquiv$ is actually a [[dependent type]]

\begin{displaymath}
f : A \to B \vdash isEquiv(f)
\end{displaymath}
on the [[function type|type of all functions]]. To obtain the [[categorical semantics]] of this general dependent $isEquiv$-construction, first notice that the interpretation of

$f : A \to B,\; a : A,\; b : B \;\vdash\; (f(a) = b)  \colon Type$

is by the rules for interpretation of [[identity types]], [[evaluation]] and [[substitution]] the left vertical morphism in the [[pullback]] [[diagram]]

\begin{displaymath}
\itexarray{  
    Q &\to&  B^I
    \\
    \downarrow && \downarrow
    \\
    [A,B] \times A \times B &\stackrel{(eval, id_B)}{\to}& B \times B
  }
  \,,
\end{displaymath}
where $eval : [A, B] \times A \to B$ is the [[evaluation map]] for the [[internal hom]]. This means that the interpretation of further [[dependent sum]] yielding $hfib$

\begin{displaymath}
f : A \to B,\; b : B 
   \; \vdash \;  
  \left(
    \sum_{a : A } (f(a) = b) 
  \right)
  \colon Type
\end{displaymath}
is the composite left vertical morphism in

\begin{displaymath}
\itexarray{  
    Q &\to& B^I
    \\
    \downarrow && \downarrow
    \\
    [A,B] \times A \times B &\stackrel{(eval, id_B)}{\to}& B \times B
    \\
    \downarrow^{\mathrlap{p_{1,3}}}
    \\
    [A,B] \times B
  }
\end{displaymath}
\end{remark}
\hypertarget{of__2}{}\subsubsection*{{Of $Equiv(-)$}}\label{of__2}

(\ldots{})

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

\begin{itemize}%
\item \textbf{isEquiv}


\item [[isContr]]


\item [[isProp]]


\item [[homotopy equivalence]]

\begin{itemize}%
\item [[weak homotopy equivalence]]


\item [[stable weak homotopy equivalence]]



\end{itemize}


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

An introduction to equivalence in homotopy type theory is in

\begin{itemize}%
\item [[Andrej Bauer]], \emph{A seminar on HoTT equivalences} (\href{http://homotopytypetheory.org/2011/12/07/a-seminar-on-hott-equivalences/}{blog post})

\end{itemize}
and basic ideas are also indicated from \href{https://home.sandiego.edu/~shulman/hottminicourse2012/02hott.pdf#page=60}{slide 60 of part 2}, \href{https://home.sandiego.edu/~shulman/hottminicourse2012/03models.pdf#page=49}{slide 49 of part 3} of

\begin{itemize}%
\item [[Mike Shulman]], \emph{Minicourse in homotopy type theory} (2012) (\href{https://home.sandiego.edu/~shulman/hottminicourse2012/}{web})

\end{itemize}
[[Coq]] code for homotopy equivalences is at

\begin{itemize}%
\item \href{https://github.com/HoTT/HoTT/blob/master/theories/Basics/Equivalences.v}{HoTT repository}

\end{itemize}
[[!redirects isEquiv]]

[[!redirects equivalence in homotopy type theory]] [[!redirects equivalences in homotopy type theory]] [[!redirects equivalence in type theory]] [[!redirects equivalences in type theory]] [[!redirects equivalence in HoTT]] [[!redirects equivalences in HoTT]]

[[!redirects adjoint equivalence in homotopy type theory]] [[!redirects adjoint equivalences in homotopy type theory]] [[!redirects adjoint equivalence in type theory]] [[!redirects adjoint equivalences in type theory]] [[!redirects adjoint equivalence in HoTT]] [[!redirects adjoint equivalences in HoTT]]

[[!redirects weak equivalence in homotopy type theory]] [[!redirects weak equivalences in homotopy type theory]] [[!redirects weak equivalence in type theory]] [[!redirects weak equivalences in type theory]] [[!redirects weak equivalence in HoTT]] [[!redirects weak equivalences in HoTT]]

[[!redirects homotopy isomorphism]] [[!redirects h-isomorphism]]



\end{document}