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\section*{higher inductive type}

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

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

[[!include type theory - contents]]

\hypertarget{induction}{}\paragraph*{{Induction}}\label{induction}

[[!include induction - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_circle}{The circle}\dotfill \pageref*{the_circle} \linebreak
\noindent\hyperlink{the_interval}{The interval}\dotfill \pageref*{the_interval} \linebreak
\noindent\hyperlink{the_2sphere}{The 2-sphere}\dotfill \pageref*{the_2sphere} \linebreak
\noindent\hyperlink{suspension}{Suspension}\dotfill \pageref*{suspension} \linebreak
\noindent\hyperlink{MappingCylinders}{Mapping cylinders}\dotfill \pageref*{MappingCylinders} \linebreak
\noindent\hyperlink{truncation}{Truncation}\dotfill \pageref*{truncation} \linebreak
\noindent\hyperlink{pushouts}{Pushouts}\dotfill \pageref*{pushouts} \linebreak
\noindent\hyperlink{QuotientsOfSets}{Quotients of sets}\dotfill \pageref*{QuotientsOfSets} \linebreak
\noindent\hyperlink{localization}{Localization}\dotfill \pageref*{localization} \linebreak
\noindent\hyperlink{spectrification}{Spectrification}\dotfill \pageref*{spectrification} \linebreak
\noindent\hyperlink{semantics}{Semantics}\dotfill \pageref*{semantics} \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}

\emph{Higher inductive types} (HITs) are a generalization of [[inductive types]] which allow the constructors to produce, not just points of the type being defined, but also elements of its iterated [[identity types]].

While HITs are already useful in [[extensional type theory]], they are most useful and powerful in [[homotopy type theory]], where they allow the construction of [[cell complexes]], [[homotopy colimits]], [[n-truncated|truncations]], [[Bousfield localization of model categories|localizations]], and many other objects from classical [[homotopy theory]].

Defining what a HIT is ``in general'' is an open research problem. One mostly precise proposal may be found in (\hyperlink{ShulmanLumsdaine16}{ShulmanLumsdaine16}). A more syntactic description of a class of HITs may be found in (\hyperlink{Brunerie16}{Brunerie16}). A solution to this problem should determine how to define the concept of an [[elementary (∞,1)-topos]].

See also [[homotopytypetheory:higher inductive type]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

All higher inductive types described below are given together with some pseudo-[[Coq]] code, which would implement that HIT if Coq supported HITs natively.

\hypertarget{the_circle}{}\subsubsection*{{The circle}}\label{the_circle}

\begin{verbatim}Inductive circle : Type :=
| base : circle
| loop : base == base.\end{verbatim}
Using the [[univalence axiom]], one can prove that the [[loop space]] {\colorbox[rgb]{1.00,0.93,1.00}{\tt base\char32\char61\char61\char32base}} of the circle type is equivalent to the [[integers]]; see \href{http://homotopytypetheory.org/2011/04/29/a-formal-proof-that-pi1s1-is-z/}{this blog post}.

\hypertarget{the_interval}{}\subsubsection*{{The interval}}\label{the_interval}

The [[homotopy type]] of the [[interval]] can be encoded as

\begin{verbatim}Inductive interval : Type :=
| zero : interval
| one : interval
| segment : zero == one.\end{verbatim}
See [[interval type]]. The interval can be proven to be [[contractible type|contractible]]. On the other hand, if the constructors {\colorbox[rgb]{1.00,0.93,1.00}{\tt zero}} and {\colorbox[rgb]{1.00,0.93,1.00}{\tt one}} satisfy their elimination rules definitionally, then the existence of an interval type implies [[function extensionality]]; see \href{http://homotopytypetheory.org/2011/04/04/an-interval-type-implies-function-extensionality/}{this blog post}. The interval can be defined as the -1-truncation of the booleans; see \href{https://groups.google.com/d/msg/homotopytypetheory/-5mLEi_qMTo/gpNUsmI-ZT4J}{here}.

\hypertarget{the_2sphere}{}\subsubsection*{{The 2-sphere}}\label{the_2sphere}

Similarly the [[homotopy type]] of the 2-[[dimensional]] [[sphere]]

\begin{verbatim}Inductive sphere2 : Type :=
| base2 : sphere2
| surf2 : idpath base2 == idpath base2.\end{verbatim}
\hypertarget{suspension}{}\subsubsection*{{Suspension}}\label{suspension}

\begin{verbatim}Inductive susp (X : Type) : Type :=
| north : susp X
| south : susp X
| merid : X -> north == south.\end{verbatim}
This is the unpointed [[suspension]]. It is also possible to define the pointed suspension. Using either one, we can define the $n$-sphere by induction on $n$, since $S^{n+1}$ is the suspension of $S^n$.

\hypertarget{MappingCylinders}{}\subsubsection*{{Mapping cylinders}}\label{MappingCylinders}

The construction of [[mapping cylinders]] is given by

\begin{verbatim}Inductive cyl {X Y : Type} (f : X -> Y) : Y -> Type :=
| cyl_base : forall y:Y, cyl f y
| cyl_top : forall x:X, cyl f (f x)
| cyl_seg : forall x:X, cyl_top x == cyl_base (f x).\end{verbatim}
Using this construction, one can define a (cofibration, trivial fibration) [[weak factorization system]] for types.

\hypertarget{truncation}{}\subsubsection*{{Truncation}}\label{truncation}

\begin{verbatim}Inductive is_inhab (A : Type) : Type :=
| inhab : A -> is_inhab A
| inhab_path : forall (x y: is_inhab A), x == y.\end{verbatim}
This is the [[(-1)-truncated|(-1)-truncation]] into [[h-propositions]]. One can prove that {\colorbox[rgb]{1.00,0.93,1.00}{\tt is\char95inhab\char32A}} is always a [[proposition]] (i.e. $(-1)$-truncated) and that it is the reflection of $A$ into propositions. More generally, one can construct the [[(effective epi, mono) factorization system]] by applying {\colorbox[rgb]{1.00,0.93,1.00}{\tt is\char95inhab}} fiberwise to a fibration.

Similarly, we have the [[0-truncated|0-truncation]] into [[h-sets]]:

\begin{verbatim}Inductive pi0 (X:Type) : Type :=
| cpnt : X -> pi0 X
| pi0_axiomK : forall (l : Circle -> pi0 X), refl (l base) == map l loop.\end{verbatim}
We can similarly define $n$-truncation for any $n$, and we should be able to define it inductively for all $n$ at once as well.

See at \emph{[[n-truncation modality]]}.

\hypertarget{pushouts}{}\subsubsection*{{Pushouts}}\label{pushouts}

The (homotopy) [[homotopy pushout|pushout]] of $f \colon A\to B$ and $g\colon A\to C$:

\begin{verbatim}Inductive hpushout {A B C : Type} (f : A -> B) (g : A -> C) : Type :=
| inl : B -> hpushout f g
| inr : C -> hpushout f g
| glue : forall (a : A), inl (f a) == inr (g a).\end{verbatim}
\hypertarget{QuotientsOfSets}{}\subsubsection*{{Quotients of sets}}\label{QuotientsOfSets}

The [[quotient]] of an [[hProp]]-value [[equivalence relation]], yielding an [[hSet]] (a [[0-truncated]] type):

\begin{verbatim}Inductive quotient (A : Type) (R : A -> A -> hProp) : Type :=
| proj : A -> quotient A R
| relate : forall (x y : A), R x y -> proj x == proj y
| contr1 : forall (x y : quotient A R) (p q : x == y), p == q.\end{verbatim}
This is already interesting in [[extensional type theory]], where [[quotient types]] are not always included. For more general homotopical quotients of ``[[internal groupoids]]'' as in the [[(∞,1)-Giraud theorem]], we first need a good definition of what such an internal groupoid is.

\hypertarget{localization}{}\subsubsection*{{Localization}}\label{localization}

Suppose we are given a family of [[function type|functions]]:

\begin{verbatim}Hypothesis I : Type.
Hypothesis S T : I -> Type.
Hypothesis f : forall i, S i -> T i.\end{verbatim}
A type is said to be $I$-\textbf{[[local object|local]]} if it sees each of these functions as an [[equivalence]]:

\begin{verbatim}Definition is_local Z := forall i,
  is_equiv (fun g : T i -> Z => g o f i).\end{verbatim}
The following HIT can be shown to be a [[reflection]] of all types into the local types, constructing the [[localization]] of the category of types at the given family of maps.

\begin{verbatim}Inductive localize X :=
| to_local : X -> localize X
| local_extend : forall (i:I) (h : S i -> localize X),
    T i -> localize X
| local_extension : forall (i:I) (h : S i -> localize X) (s : S i),
    local_extend i h (f i s) == h s
| local_unextension : forall (i:I) (g : T i -> localize X) (t : T i),
    local_extend i (g o f i) t == g t
| local_triangle : forall (i:I) (g : T i -> localize X) (s : S i),
    local_unextension i g (f i s) == local_extension i (g o f i) s.\end{verbatim}
The first constructor gives a map from {\colorbox[rgb]{1.00,0.93,1.00}{\tt X}} to {\colorbox[rgb]{1.00,0.93,1.00}{\tt localize\char32X}}, while the other four specify exactly that {\colorbox[rgb]{1.00,0.93,1.00}{\tt localize\char32X}} is local (by giving adjoint equivalence data to the map that we want to become an equivalence). See \href{http://homotopytypetheory.org/2011/12/06/inductive-localization/}{this blog post} for details. This construction is also already interesting in extensional type theory.

\hypertarget{spectrification}{}\subsubsection*{{Spectrification}}\label{spectrification}

A [[prespectrum]] is a sequence of [[pointed object|pointed types]] $X_n$ with pointed maps $X_n \to \Omega X_{n+1}$:

\begin{verbatim}Definition prespectrum :=
  {X : nat -> Type & 
   { pt : forall n, X n &
    { glue : forall n, X n -> pt (S n) == pt (S n) &
      forall n, glue n (pt n) == idpath (pt (S n)) }}}.\end{verbatim}
A prespectrum is a [[spectrum]] if each of these maps is an equivalence.

\begin{verbatim}Definition is_spectrum (X : prespectrum) : Type :=
  forall n, is_equiv (pr1 (pr2 (pr2 X)) n).\end{verbatim}
In classical [[algebraic topology]], there is a \textbf{spectrification} functor which is [[left adjoint]] to the inclusion of spectra in prespectra. For instance, this is how a [[suspension spectrum]] is constructed: by spectrifying the prespectrum $X_n \coloneqq \Sigma^n A$.

The following HIT should construct spectrification in [[homotopy type theory]] (though this has not yet been verified formally). (There are some abuses of notation below, which can be made precise using Coq typeclasses and implicit arguments.)

\begin{verbatim}Inductive spectrify (X : prespectrum) : nat -> Type :=
| to_spectrify : forall n, X n -> spectrify X n
| spectrify_glue : forall n, spectrify X n ->
    to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))
| to_spectrify_is_prespectrum_map : forall n (x : X n),
    spectrify_glue n (to_spectrify n x)
    == loop_functor (to_spectrify (S n)) (glue n x)
| spectrify_glue_retraction : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify X n
| spectrify_glue_retraction_is_retraction : forall n (sx : spectrify X n),
    spectrify_glue_retraction n (spectrify_glue n sx) == sx
| spectrify_glue_section : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify X n
| spectrify_glue_section_is_section : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify_glue n (spectrify_glue_section n p) == p.\end{verbatim}
We can unravel this as follows, using more traditional notation. Let $L X$ denote the spectrification being constructed. The first constructor says that each $(L X)_n$ comes with a map from $X_n$, called $\ell_n$ say (denoted {\colorbox[rgb]{1.00,0.93,1.00}{\tt to\char95spectrify\char32n}} above). This induces a basepoint in each type $(L X)_n$, namely the image $\ell_n(*)$ of the basepoint of $X_n$. The many occurrences of

\begin{verbatim}to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))\end{verbatim}
simply refer to the based \emph{loop space} of $\Omega_{\ell_{n+1}(*)} (L X)_{n+1}$ of $(L X)_{n+1}$ at this base point.

Thus, the second constructor {\colorbox[rgb]{1.00,0.93,1.00}{\tt spectrify\char95glue}} gives the structure maps $(L X)_n \to \Omega (L X)_{n+1}$ to make $L X$ into a prespectrum. Similarly, the third constructor says that the maps $\ell_n\colon X_n \to (L X)_n$ commute with the structure maps up to a specified homotopy.

Since the basepoints of the types $(L X)_n$ are induced from those of each $X_n$, this automatically implies that the maps $(L X)_n \to \Omega (L X)_{n+1}$ are pointed maps (up to a specified homotopy) and that the $\ell_n$ commute with these pointings (up to a specified homotopy). This makes $\ell$ into a map of prespectra.

Finally, the fourth through seventh constructors say that $L X$ is a spectrum, by giving [[h-isomorphism]] data: a retraction and a section for each glue map $(L X)_n \to \Omega (L X)_{n+1}$. We could use adjoint equivalence data as we did for localization, but this approach avoids the presence of level-3 path constructors. (We could have used h-iso data in localization too, thereby avoiding even level-2 constructors there.) It is important, in general, to use a sort of equivalence data which forms an [[h-prop]]; otherwise we would be adding [[stuff, structure, property|structure]] rather than merely the property of such-and-such map being an equivalence.

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

See (\hyperlink{LumsdaineShulman17}{Lumsdaine-Shulman17}).

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

\begin{itemize}%
\item [[inductive type]], [[initial algebra of an endofunctor]]


\item \textbf{higher inductive type}, [[initial algebra of a presentable ∞-monad]]



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

Expositions include

\begin{itemize}%
\item [[Mike Shulman]], \emph{Homotopy type theory IV} (\href{http://golem.ph.utexas.edu/category/2011/04/homotopy_type_theory_vi.html}{blog entry})


\item [[Peter LeFanu Lumsdaine]], \emph{Higher inductive types, a tour of the menageries} (\href{http://homotopytypetheory.org/2011/04/24/higher-inductive-types-a-tour-of-the-menagerie/}{blog post})


\item [[Peter LeFanu Lumsdaine]], \emph{Higher Inductive Types: The circle and friends, axiomatically} (\href{http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS2011/Lumsdaine_slides.pdf}{pdf})



\end{itemize}
Details of the semantics are in

\begin{itemize}%
\item [[Peter LeFanu Lumsdaine]] [[Mike Shulman]], \emph{Semantics of higher inductive types} (\href{https://arxiv.org/abs/1705.07088}{arXiv:1705.07088}, talk slides \href{http://home.sandiego.edu/~shulman/papers/cellcxs.pdf}{pdf})

\end{itemize}
with precursors in

\begin{itemize}%
\item [[Mike Shulman]], [[Peter LeFanu Lumsdaine]], \emph{Semantics of higher inductive types}, 2012 (\href{http://uf-ias-2012.wikispaces.com/file/view/semantics.pdf/410646692/semantics.pdf}{pdf})


\item [[Mike Shulman]], [[Peter LeFanu Lumsdaine]], \emph{Semantics and syntax of higher inductive types}, 2016, (\href{http://home.sandiego.edu/~shulman/papers/stthits.pdf}{slides})



\end{itemize}
Discussion of a subset of the HITs is in:

\begin{itemize}%
\item [[Kristina Sojakova]], \emph{Higher Inductive Types as Homotopy-Initial Algebras} \href{http://arxiv.org/abs/1402.0761}{arXiv:1402.0761}


\item [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], \emph{Homotopy-initial algebras in type theory} (\href{http://arxiv.org/abs/1504.05531}{arXiv:1504.05531})


\item [[Michael Rathjen]], \emph{\href{http://www2.macs.hw.ac.uk/~cm389/hexmaps/2014/03/epsrc-ict-50/grants/EP-K023128-1.php}{Homotopical Inductive Types}} on [[higher inductive types]]



\end{itemize}
Implementation in [[Agda]]/[[Coq]] is discussed in

\begin{itemize}%
\item [[Guillaume Brunerie]], \emph{Implementation of higher inductive types in HoTT-Agda}, 2016, \href{https://github.com/HoTT/HoTT-Agda/blob/master/core/lib/types/HIT_README.txt}{github}


\item Bruno Barras, \emph{Native implementation of Higher Inductive Types (HITs) in Coq} \href{http://www.crm.cat/en/activities/documents/barras-crm-2013.pdf}{PDF}



\end{itemize}
[[!redirects higher inductive types]]

[[!redirects HIT]] [[!redirects HITs]]



\end{document}