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\begin{document}

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\section*{inductive type}

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

\hypertarget{deduction_and_induction}{}\paragraph*{{Deduction and Induction}}\label{deduction_and_induction}

[[!include deduction and induction - contents]]

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{InductionRules}{Induction: dependent elimination, computation}\dotfill \pageref*{InductionRules} \linebreak
\noindent\hyperlink{RecursionRules}{Recursion: elimination, computation}\dotfill \pageref*{RecursionRules} \linebreak
\noindent\hyperlink{categorical_semantics}{Categorical semantics}\dotfill \pageref*{categorical_semantics} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{homotopy_initiality}{Homotopy initiality}\dotfill \pageref*{homotopy_initiality} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{NaturalNumbers}{Natural numbers}\dotfill \pageref*{NaturalNumbers} \linebreak
\noindent\hyperlink{introduction_elimination_computation}{Introduction, elimination, computation}\dotfill \pageref*{introduction_elimination_computation} \linebreak
\noindent\hyperlink{induction_2}{Induction}\dotfill \pageref*{induction_2} \linebreak
\noindent\hyperlink{recursion}{Recursion}\dotfill \pageref*{recursion} \linebreak
\noindent\hyperlink{IdentityTypes}{Identity types}\dotfill \pageref*{IdentityTypes} \linebreak
\noindent\hyperlink{categorical_semantics_2}{Categorical semantics}\dotfill \pageref*{categorical_semantics_2} \linebreak
\noindent\hyperlink{PathInduction}{Path induction}\dotfill \pageref*{PathInduction} \linebreak
\noindent\hyperlink{PathRecursion}{Path recursion}\dotfill \pageref*{PathRecursion} \linebreak
\noindent\hyperlink{wtypes}{W-types}\dotfill \pageref*{wtypes} \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}

An \emph{inductive type} is\ldots{}

In terms of [[categorical semantics]], an \emph{inductive type} is a type whose interpretation is given by an [[initial algebra of an endofunctor]].

This has the usual meaning in ordinary [[category theory]]. In applications to [[(∞,1)-category theory]], the uniqueness clause in the notion of initial object is modified to allow for a [[contractible space]] of choices (as discussed at \emph{[[initial object in an (∞,1)-category]]}), and this difference is reflected accordingly in the type-theoretic set-up. The [[syntax]] will give back the traditional meaning whenever equality is interpreted [[extensional type theory|extensionally]].

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

There are two \emph{equivalent} ways of defining the [[judgement]] rules for inductive types. The first describes [[term elimination|elimination]] on [[dependent types]] over the given type. This is the formalization of the notion of [[induction]], and discussed below in

\begin{itemize}%
\item \hyperlink{InductionRules}{Induction: dependent elimination, computation}.

\end{itemize}
The second describes [[term elimination|elimination]] on absolute types. This is the formalization of the notion of [[recursion]], and discussed below

\begin{itemize}%
\item \hyperlink{RecursionRules}{Recursion: elimination, computation}

\end{itemize}
\hypertarget{InductionRules}{}\subsubsection*{{Induction: dependent elimination, computation}}\label{InductionRules}

(\ldots{})

\hypertarget{RecursionRules}{}\subsubsection*{{Recursion: elimination, computation}}\label{RecursionRules}

(\ldots{})

\hypertarget{categorical_semantics}{}\subsubsection*{{Categorical semantics}}\label{categorical_semantics}

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

\begin{defn}
\label{InterpretationOfTheRules}\hypertarget{InterpretationOfTheRules}{}
The categorical interpretation of [[induction]], hence of the \emph{dependent elimination} and computation rules from \hyperlink{InductionRules}{above} are the following.

Let $\mathcal{C}$ be the [[ambient category]].

\begin{enumerate}%
\item \textbf{[[term introduction rule]]}

The interpretation of inductive term introduction is by an [[endofunctor]] $F : \mathcal{C} \to \mathcal{C}$ and an [[algebra over an endofunctor]], exhibited by a [[morphism]] in $\mathcal{C}$ of the form

\begin{displaymath}
F(W) \to W
  \,.
\end{displaymath}

\item \textbf{[[term elimination rule]]}

The interpretation of the dependent elimination rule says that given a [[display map]] $B \to W$, where $B$ is given an $F$-[[algebra over an endofunctor|algebra structure]] and the display map is an $F$-algebra [[homomorphism]], the dependent eliminator is interpreted as a specified [[section]] $\sigma : W \to B \in \mathcal{C}_{/W}$, hence as a [[diagram]]

\begin{displaymath}
\itexarray{
    W &&\stackrel{\sigma}{\to}&& B
    \\
     & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{}}
    \\
    && W
  }
\end{displaymath}
in $\mathcal{C}$.


\item \textbf{[[computation rule]]}

The interpretation of the dependent computation rules is that the section $\sigma$ from above is required to be an [[algebra for an endofunctor|algebra]] [[homomorphism]].



\end{enumerate}
\end{defn}
\begin{defn}
\label{InterpretationOfTheSimpleRules}\hypertarget{InterpretationOfTheSimpleRules}{}
The categorical interpretation of [[recursion]], hence of the absolute elimination rules from \hyperlink{RecursionRules}{above} in a suitable [[category]] $\mathcal{C}$ is the following

\begin{enumerate}%
\item \textbf{[[term introduction rule]]}

The interpretation of inductive term introduction is by an [[endofunctor]] $F : \mathcal{C} \to \mathcal{C}$ and an [[algebra over an endofunctor]], exhibited by a [[morphism]] in $\mathcal{C}$ of the form

\begin{displaymath}
F(W) \to W
  \,.
\end{displaymath}

\item \textbf{[[term elimination rule]]}

The interpretation of the absolute elimination rule is that for $A$ any other $F$-[[algebra of an endofunctor|algebra]], there is a morphism $W \to A$ in $\mathcal{C}$.


\item \textbf{[[computation rule]]}

The interpretation of the absolute computation rule says that the morphism $W \to A$ from above is an algebra [[homomorphism]] and is unique as such.



\end{enumerate}
In summary this says that the recursion rules are interpreted as an [[initial algebra of an endofunctor]].

\end{defn}
\begin{prop}
\label{BothFormulationsOfInitialityAreEquivalent}\hypertarget{BothFormulationsOfInitialityAreEquivalent}{}
When interpreted in a category $\mathcal{C}$ of [[homotopy type|homotopy 0-types]] = sets, definition \ref{InterpretationOfTheRules} and definition \ref{InterpretationOfTheSimpleRules} are indeed equivalent.

\end{prop}
\begin{proof}
First suppose that $W$ is an initial $F$-algebra as in def. \ref{InterpretationOfTheSimpleRules}. Then since [[initial object|initiality]] entails first of all the existence of a morphims to any other object it follows that with $B$ another $F$-algebra there is a homomorphism $W \to B$, and since secondly initiality entails uniqueness of this morphism, it follows that given a homomorphism $B \to W$ the composite $W \to B \to W$ has to equal the identity $id_W$, hence that $B \to W$ has a section, and uniquely so.

Conversely, assume that $W$ satisfies definition \ref{InterpretationOfTheRules}. For $A$ any other $F$-algebra we can form the trivial display map $W \times A \to W$ by [[projection]] and a [[section]] of this is equivalently just a morphism $W \to A$, so we have a homomorphism from $W$ to any other $F$-algebra $A$. Therefore to show that $W$ is an initial $F$-algebra it remains to show that for $f, g : W \to A$ two algebra homomorphism, they are necessarily equal.

To that end, notice that by the assumption of 0-truncation, the [[diagonal]] $\delta \colon A \to A \times A$ is a display map / fibration.

Form its [[pullback]] $P$

\begin{displaymath}
\itexarray{
    P & \to & A 
    \\
    \downarrow & & \downarrow^\mathrlap{\delta} 
    \\
    W & \underset{\langle f, g \rangle}{\to} & A \times A,
  }
  \,,
\end{displaymath}
which is also an algebra homomorphism. Therefore by the interpretation of the elimination rule it has a (specified) [[section]] $\sigma : W \to P$. But $P \to W$ is the pullback of a [[monomorphism]] and therefore itself a monomorphism, and so the section forces it to be in fact an [[isomorphism]]. This in turn means that $f$ and $g$ are equal.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
In [[intensional type theory]], where the diagonal is not a display map, we can perform the same argument using a [[path object]] $P A \to A \times A$ (represented in type theory by an [[identity type]]), showing thereby that $f$ and $g$ are homotopic. A fancier version of this argument enables us to show that the space of algebra maps $W\to A$ is actually contractible. In other words, the axioms for an inductive type still imply that algebra maps out of $W$ are essentially unique, even though the axioms do not state this explicitly.

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

\hypertarget{homotopy_initiality}{}\subsubsection*{{Homotopy initiality}}\label{homotopy_initiality}

Any inductive type $W$ is a homotopy initial F-[[algebra over an endofunctor|algebra]]: the space of $F$-[[algebra for an endofunctor|algebra]] maps $W \to X$ is [[contractible]].

(\hyperlink{AwodeyGambinoSojakova}{Awodey-Gambino-Sojakova})

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

\hypertarget{NaturalNumbers}{}\subsubsection*{{Natural numbers}}\label{NaturalNumbers}

\hypertarget{introduction_elimination_computation}{}\paragraph*{{Introduction, elimination, computation}}\label{introduction_elimination_computation}

\begin{defn}
\label{InterpretationOfTheRules}\hypertarget{InterpretationOfTheRules}{}
The [[type of natural numbers]] $\mathbb{N}$ is the inductive type defined as follows.

\begin{enumerate}%
\item \textbf{[[term introduction rule]]}

\begin{displaymath}
\frac{}{0 \in \mathbb{N}}
  \;\;
  \frac{n \in \mathbb{N}}{s(n) \in \mathbb{N}}
\end{displaymath}

\item \textbf{[[term elimination rule]]}

\begin{displaymath}
\frac{
     \left(
     x \in \mathbb{N} \vdash  P(x) : Type
     \;\;
     \vdash p_0 : P(0)
     \;\;\;
     x \in \mathbb{N}, p : P(x) \vdash q : P(s (x))
    \right)
    \;\;
    \vdash n \in \mathbb{N}
  }{
    p(n) : P(n)
  }
\end{displaymath}
(check, this probably still has syntax errors\ldots{})


\item \textbf{[[computation rule]]}

(\ldots{})



\end{enumerate}
\end{defn}
See for instance (\hyperlink{Pfenning}{Pfenning, section 2}).

In [[Coq]]-[[syntax]] the [[natural numbers]] are the inductive type defined by

\begin{verbatim}Inductive nat : Type :=
 | zero : nat
 | succ : nat -> nat.\end{verbatim}
In the [[categorical semantics]] this is interpreted as the [[initial algebra for an endofunctor|initial algebra]] for the endofunctor $F$ that sends an object to its [[coproduct]] with the [[terminal object]]

\begin{displaymath}
F(X) = * \coprod X
  \,,
\end{displaymath}
or in different equivalent notation, which is very suggestive here:

\begin{displaymath}
F(X) = 1 + X
  \,.
\end{displaymath}
That [[initial algebra for an endofunctor|initial algebra]] is (as explained there) precisely a [[natural number object]] $\mathbb{N}$. The two components of the morphism $F(\mathbb{N}) \to \mathbb{N}$ that defines the algebra structure are the 0-[[generalized element|element]] $zero : * \to \mathbb{N}$ and the successor [[endomorphism]] $successor : \mathbb{N} \to \mathbb{N}$

\begin{displaymath}
(zero,successor) : * \coprod \mathbb{N} \to \mathbb{N}
  \,.
\end{displaymath}
In the following we write of course for short $0 : * \to \mathbb{N}$ and $s : \mathbb{N} \to \mathbb{N}$.

\hypertarget{induction_2}{}\paragraph*{{Induction}}\label{induction_2}

\begin{example}
\label{}\hypertarget{}{}
We spell out in detail how the fact that $\mathbb{N}$ satisfied def. \ref{InterpretationOfTheRules} is the classical [[induction principle]].

That principle says informally that if a [[proposition]] $P$ depending on the natural numbers is true at $n = 0$ and such that if it is true for some $n$ then it is true for $n+1$, then it is true for all natural numbers.

Here is how this is formalized in type theory and then [[categorical semantics|interpreted]] in some suitable ambient category $\mathcal{C}$.

First of all, that $P$ is a proposition depending on the natural numbers means that it is a [[dependent type]]

\begin{displaymath}
n \in \mathbb{N} \vdash P(n) : Type
  \,.
\end{displaymath}
The categorical interpretation of this is by a [[display map]]

\begin{displaymath}
\itexarray{
     P 
     \\
     \downarrow
     \\
     \mathbb{N}
  }
\end{displaymath}
in the given category $\mathcal{C}$.

Next, the fact that $P$ holds at 0 means that there is a ([[proof]]-)[[term]]

\begin{displaymath}
\vdash p_0 \in P(0)
  \,.
\end{displaymath}
In the categorical semantics the [[substitution]] of $n$ for 0 that gives $P(0)$ is given by the [[pullback]] of the above fibration

\begin{displaymath}
\itexarray{
    0^* P &\to& P
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{0}{\to}& \mathbb{N}
  }
\end{displaymath}
and the [[term]] $p_0$ is interpreted as a [[section]] of the resultig fibration over the terminal object

\begin{displaymath}
\itexarray{
    * &\stackrel{p_0}{\to}& 0^* P &\to& P
    \\
    &\searrow& \downarrow && \downarrow
    \\
    && * &\stackrel{0}{\to}& \mathbb{N}
  }
  \,.
\end{displaymath}
But by the defining [[universal property]] of the pullback, this is equivalently just a [[commuting diagram]]

\begin{displaymath}
\itexarray{
    * &\stackrel{p_0}{\to}& P
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{0}{\to}& \mathbb{N}
  }
  \,.
\end{displaymath}
Next the induction step. Formally it says that for all $n \in \mathbb{N}$ there is an [[implication]] $p_s(n) : P(n) \to P(n+1)$

\begin{displaymath}
n \in \mathbb{N} \vdash p_s(n)  : P(n) \to   P(n+1)
\end{displaymath}
The categorical semantics of the [[substitution]] of $n+1$ for $n$ is now given by the [[pullback]]

\begin{displaymath}
\itexarray{
     P((-)+1) \coloneqq & s^*P &\to& P
     \\
     & \downarrow && \downarrow 
     \\
     & \mathbb{N} &\stackrel{s}{\to}& \mathbb{N}
  }
\end{displaymath}
and the interpretation of the implication term $p_s(n)$ is as a morphism $P \to s^* P$ in $\mathcal{C}_{/\mathbb{N}}$

\begin{displaymath}
\itexarray{
     P & \stackrel{p_s}{\to} &  s^*P &\to& P
     \\
     &\searrow & \downarrow && \downarrow 
     \\
     && \mathbb{N} &\stackrel{s}{\to}& \mathbb{N}
  }
  \,.
\end{displaymath}
Again by the [[universal property]] of the pullback this is the same as a [[commuting diagram]]

\begin{displaymath}
\itexarray{
     P &\stackrel{p_s}{\to}& P
     \\
     \downarrow && \downarrow 
     \\
     \mathbb{N} &\stackrel{s}{\to}& \mathbb{N}
  }
  \,.
\end{displaymath}
In summary this shows that the fact that $P$ is a proposition depending on natural numbers which holds at 0 and which holds at $n+1$ if it holds at $n$ is interpreted precisely as an $F$-[[algebra for an endofunctor|algebra homomorphism]]

\begin{displaymath}
\itexarray{
     P
     \\
     \downarrow
     \\
     \mathbb{N}
  }
  \,.
\end{displaymath}
The [[induction principle]] is supposed to deduce from this that $P$ holds for every $n$, hence that there is a proof $p_n : P(n)$ for all $n$:

\begin{displaymath}
n \in \mathbb{N} \vdash p(n) : P(n)
  \,.
\end{displaymath}
The categorical interpretation of this is as a morphism $p : \mathbb{N} \to P$ in $\mathcal{C}_{/\mathbb{N}}$. The existence of this is indeed exactly what the interpretation of the elimination rule, def. \ref{InterpretationOfTheRules}, gives, or (equivalently by prop. \ref{BothFormulationsOfInitialityAreEquivalent}) exactly what the initiality of the $F$-algebra $\mathbb{N}$ gives.

\end{example}
\hypertarget{recursion}{}\paragraph*{{Recursion}}\label{recursion}

\begin{example}
\label{}\hypertarget{}{}
We spell out how the fact that $\mathbb{N}$ satisfies def. \ref{InterpretationOfTheSimpleRules} is the classical [[recursion principle]].

So let $A$ be an $F$-algebra object, hence equipped with a morphism

\begin{displaymath}
a_0 : * \to A
\end{displaymath}
and a morphism

\begin{displaymath}
h : A \to A
  \,.
\end{displaymath}
By [[initial object|initiality]] of the $F$-algebra $\mathbb{N}$, there is then a (unique) morphism

\begin{displaymath}
f : \mathbb{N} \to A
\end{displaymath}
such that the diagram

\begin{displaymath}
\itexarray{
    * &\stackrel{0}{\to}& \mathbb{N} &\stackrel{(-)+1}{\to}& \mathbb{N}
    \\
    \downarrow && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f}}
    \\
    * &\stackrel{a_0}{\to}& A &\stackrel{h}{\to}& A
  }
\end{displaymath}
commutes. This means precisely that $f$ is the function defined recursively by

\begin{enumerate}%
\item $f(0) = a_0$;


\item $f(n+1) = h(f(n))$.



\end{enumerate}
\end{example}
\hypertarget{IdentityTypes}{}\subsubsection*{{Identity types}}\label{IdentityTypes}

The introduction, elimination and computation rules for \emph{[[identity types]]} are discussed there.

In [[Coq]]-[[syntax]] the [[identity types]] are the inductive types (or more precisely, the \emph{[[inductive family]]}) defined by

\begin{verbatim}Inductive id {A} : A -> A -> Type := 
  idpath : forall x, id x x.\end{verbatim}
\hypertarget{categorical_semantics_2}{}\paragraph*{{Categorical semantics}}\label{categorical_semantics_2}

We may [[categorical semantics|interpret]] identity types in suitable categories $\mathcal{C}$ such as a [[type-theoretic model category]].

\begin{example}
\label{EndofunctorForIdentityTypes}\hypertarget{EndofunctorForIdentityTypes}{}
The [[categorical semantics|categorical interpretation]] of identity types in a category $\mathcal{C}$ is as the [[initial algebra of an endofunctor|initial algebra]] for the [[endofunctor]]

\begin{displaymath}
F : \mathcal{C}_{/A \times A} \to 
  \mathcal{C}_{/A \times A}
\end{displaymath}
of the [[slice category]] $\mathcal{C}_{/A \times A}$ over $A\times A$ which is constant at the [[diagonal]] $A\to A\times A$:

\begin{displaymath}
F (\langle E \to A \times A\rangle) = \langle A \stackrel{\Delta}{\to} A \times A\rangle
  \,.
\end{displaymath}
\end{example}
So an [[algebra for an endofunctor|algebra]] for this endofunctor is a morphism

\begin{displaymath}
\itexarray{
    A &&\to&& E
    \\
    & {}_{\mathllap{\Delta}}\searrow && \swarrow
    \\
    && A \times A
  }
\end{displaymath}
and the [[initial object|initial]] such is the [[path space object]] $A^I \to A \times A$.

\hypertarget{PathInduction}{}\paragraph*{{Path induction}}\label{PathInduction}

\begin{example}
\label{}\hypertarget{}{}
We spell out in detail how the the [[induction principle]] def. \ref{InterpretationOfTheRules} for identity types is the [[principle of substitution of equals for equals]].

To have an $F$-algebra $\langle E \to A\times A\rangle$ over $\langle A^I \to A \times A\rangle$ means precisely to have a diagram

\begin{displaymath}
\itexarray{
    && E 
    \\
    & \nearrow& \downarrow
    \\
    A &\to& A^I
    \\
    &\searrow& \downarrow
    \\
    && A \times A
  }
\end{displaymath}
in $\mathcal{C}$.

This is the interpretation of the elimination rule: $E \to A^I$ is the interpretation of a type

\begin{displaymath}
a,b \in A , p : (a = b) \vdash E(a,b,p)
\end{displaymath}
and the lift $A \to E$ is a [[section]] of the [[pullback]] of $E$ to $A$, hence an interpretation of a [[term]] in the [[substitution]]

\begin{displaymath}
s : E(a,a,r_a)
  \,.
\end{displaymath}
The elimination rule then says that this extends to a section $A^I \to E$, hence a ``proof of $E$ over all identifications'' $a = b$.

\end{example}
\hypertarget{PathRecursion}{}\paragraph*{{Path recursion}}\label{PathRecursion}

\begin{example}
\label{}\hypertarget{}{}
We spell out how the the [[recursion principle]] def. \ref{InterpretationOfTheSimpleRules} for [[identity types]] is related to the \emph{[[complete Segal space|Segal-completeness condition]]} and in particular to \emph{[[univalence]]}.

Notice that an [[algebra for an endofunctor|algebra over the endofunctor]] that defines identity types, example \ref{EndofunctorForIdentityTypes},

\begin{displaymath}
\itexarray{
     X_0 &&\stackrel{\sigma_0}{\to}&& X_1
     \\
     & \searrow && \swarrow_{\mathrlap{\delta_0, \delta_1}}
     \\
     && X_0 \times X_0
  }
\end{displaymath}
constitutes the [[simplicial skeleton|1-skeleton]] of a [[simplicial object]]

\begin{displaymath}
\itexarray{
     X_1
     \\
     {}^{\mathllap{\delta_0}}\downarrow \uparrow^{\mathrlap{\sigma_0}} \downarrow^{\mathrlap{\delta_1}}
     \\   
     X_0
  }
  \,.
\end{displaymath}
The [[recursion principle]] says that the [[simplicial identities|degeneracy map]] $\sigma_0$ factors through the [[path space object]] of $X_0$ as a lift $\hat \sigma_0$ in the diagram

\begin{displaymath}
\itexarray{
     X_0 &\stackrel{\sigma_0}{\to}& X_1
     \\
     \downarrow &\nearrow_{\hat \sigma_0}& \downarrow
     \\
     X_0^I &\to& X_0 \times X_0
  }
  \,.
\end{displaymath}
[[categorical semantics|Semantically]], this lift exists because $X_0 \to X_0^I$ is an [[acyclic cofibration]] by definition of [[path space object]], and $X_1 \to X_0 \times X_0$ is a [[fibration]] ([[display map]]) by the interpretation rule for [[dependent types]].

This morphism

\begin{displaymath}
\hat \sigma_0 : X_0^I \to X_1
\end{displaymath}
lifts paths/[[morphisms]] that exist in $X_0$ to the morphisms exhibited by $X_1$, if we think of the above as the 1-skeleton of a simplicial object that represents an [[internal category in an (infinity,1)-category]].

Suppose this exists, then there will be a notion of \emph{equivalences} in $X_1$, those morphisms that are invertible with respect to the given composition operation. In good situations this will give the [[core]] inclusion

\begin{displaymath}
Core(X_1) \hookrightarrow X_1
  \,.
\end{displaymath}
In this case the [[complete Segal space|Segal-completeness condition]] in degree 1 says that the path recursion $\hat \sigma_0$ exhibits this inclusion

\begin{displaymath}
\hat \sigma_0 : X_0^I \simeq Core(X_1) \to X_1
  \,.
\end{displaymath}
In the case that $X_\bullet$ is the classifier of the [[codomain fibration]], then this is called the \emph{[[univalence]]-condition}.

\end{example}
\hypertarget{wtypes}{}\subsubsection*{{W-types}}\label{wtypes}

\begin{itemize}%
\item [[W-type]]

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

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


\item [[inductive-inductive type]]


\item [[inductive family]]


\item [[positive type]], [[negative type]]


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


\item [[coinductive type]]



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

A very basic introduction to the concept, with an eye towards explaining [[identity types]] is in

\begin{itemize}%
\item [[Mike Shulman]], \emph{Induction on equality} (\href{http://home.sandiego.edu/~shulman/papers/induction.pdf}{pdf})

\end{itemize}
A textbook account in the context of [[programming languages]] is in section 15 of

\begin{itemize}%
\item [[Robert Harper]], \emph{[[Practical Foundations for Programming Languages]]}

\end{itemize}
Discussion of inductive types in the context of [[Coq]]-programming is in chapter 3 of

\begin{itemize}%
\item [[Adam Chlipala]], \emph{\href{http://adam.chlipala.net/cpdt/}{Certified programming with dependent types}}

\end{itemize}
See also

\begin{itemize}%
\item Michael Abbott, [[Thorsten Altenkirch]], [[Neil Ghani]], \emph{Inductive Types for Free -- Representing nested inductive types using W-types} (\href{http://www.cs.nott.ac.uk/~txa/talks/icalp-slides.pdf}{pdf})

\end{itemize}
Expositions with an eye towards [[higher inductive types]] 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}{web})


\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 [[Mike Shulman]], \emph{Inductive and higher inductive types}, talk slides (2012) (\href{http://www.sandiego.edu/~shulman/hottminicourse2012/04induction.pdf}{pdf})



\end{itemize}
Original references include

\begin{itemize}%
\item [[Per Martin-Löf]], \emph{Intuitionistic Type Theory}. Notes by G. Sambin of a series of lectures given in Padua, 1980. Bibliopolis, 1984.

\end{itemize}
The formalization in [[Coq]] is discussed in

\begin{itemize}%
\item Eduardo Gim\'e{}nez, Pierre Cast\'e{}ran, \emph{A Tutorial on Co-Inductive Types in Coq} (\href{http://flint.cs.yale.edu/cs428/coq/pdf/RecTutorial.pdf}{pdf})

\end{itemize}
A study of the homotopy-initiality of inductive types in [[homotopy type theory]] is in

\begin{itemize}%
\item [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], \emph{Inductive types in homotopy type theory} (\href{http://arxiv.org/abs/1201.3898}{arXiv:1201.3898})

\end{itemize}
The corresponding [[Coq]]-code is at

\begin{itemize}%
\item \href{https://github.com/HoTT/HoTT/tree/master/Coq/IT}{https://github.com/HoTT/HoTT/tree/master/Coq/IT}

\end{itemize}
Exposition and discussion of that result is in

\begin{itemize}%
\item [[Steve Awodey]], \emph{Inductive types in Hott} (\href{http://homotopytypetheory.org/2012/01/19/inductive-types-in-hott/}{blog post})

\end{itemize}
Discussion of the inductive type of [[natural numbers]] is in

\begin{itemize}%
\item Frank Pfenning, \emph{Lecture notes on natural numbers} (2009) (\href{http://www.cs.cmu.edu/~fp/courses/15317-f09/lectures/06-nat.pdf}{pdf})

\end{itemize}
Discussion of inductive types in the context of [[linear type theory]] is in

\begin{itemize}%
\item St\'e{}phane Gimenez, \emph{Towards Generic Inductive Constructions in Systems of Nets} (\href{http://www.imn.htwk-leipzig.de/WST2013/papers/paper_16.pdf}{pdf})

\end{itemize}
[[!redirects inductive type]] [[!redirects inductive types]] [[!redirects inducive type]]



\end{document}