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\begin{document}

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\section*{higher inductive type}

1Categorially a \emph{higher inductive type} in an extensional type theory is an initial algebra of an endofunctor. In intensional type theory this analogy fails.

In particular $W$[[W-type|-type]]s in an extensional type theory correspond to initial algebras of [[nLab:polynomial functor]]s. Also this is not true for intensional type theories.

This failure of intensional type theory can be (at least for $W$-types and some more general cases) remedied by replacing ``initial'' by ``homotopy initial''. This is the main result (to be found in §2) of

\begin{itemize}%
\item Steve Awodey, Nicola gambino, Kristina Sojakova, inductive types in homotopy type theory, \href{http://arxiv.org/abs/1201.3898}{arXiv:1201.3898}

\end{itemize}
\hypertarget{extensional_vs_intensional_type_theories}{}\subsection*{{Extensional vs. intensional type theories}}\label{extensional_vs_intensional_type_theories}

The four standard forms of typing judgements are

(1) $A:Type$

(2) $A=B:Type$ \emph{definitional equality for types}

(3) $a:A$

(4) $a=b:A$ \emph{definitional equality for termes}

There is also another notion of equality - namely \emph{propositional equality}:

\hypertarget{rules_for_identity_types}{}\subsubsection*{{Rules for identity types}}\label{rules_for_identity_types}

(1) $Id$-formation rule

\begin{displaymath}
\frac{A:Type\; a:A\; b:A}{Id_A(a,b):Type}
\end{displaymath}
(2) $Id$-introduction rule

\begin{displaymath}
\frac{a:A}{refl(a):Id_A(a,a)}
\end{displaymath}
(3) $Id$-elimination rule

\begin{displaymath}
\frac{\itexarray{a,y:A,u:Id_A(x,y)\vdash C(x,y,u):Type\\x:A\vdash c(x):C(x,x,refl(x))}}{x,y:A,u:Id_A(x,y)\vdash idrec(x,y,u,c):C(x,y,u)}
\end{displaymath}
(4) $Id$-computation rule

\begin{displaymath}
\frac{\itexarray{a,y:A,u:Id_A(x,y)\vdash C(x,y,u):Type\\x:A\vdash c(x):C(x,x,refl(x))}}{x:A\vdash idrec(x,x,refl(x),c)=c(x):C(x,xrefl(x))}
\end{displaymath}
One important effect of not having the \emph{identity-reflection rule}

\begin{displaymath}
\frac{p:Id_A(x,y)}{x=y:A}
\end{displaymath}
is that it is impossible to prove that the empty type is an initial object. There are some \emph{extensionality principles} which are weaker than the identity reflection rule: Streicher's K-rule, the \emph{Uniqueness of Identity Proofs} (UIP) which also has benn considered in context of Observational Type Theory. However these constructions seem to be ad hoc and lack a conceptual basis.

\hypertarget{the_system_}{}\subsection*{{The system $\mathcal{H}$}}\label{the_system_}

The dependent type theory $\mathcal{H}$ is defined to have -in addition to the [[rules of type theories|standard structural rules]]- the folowing rules:

(1) the rules for identity types.

(2) rules for $\Sigma$-types as presented in Nordstrom-Petterson-Smith §5.8

(3) rules for $\Pi$-types as presented in Garner §3.2

(4) the propositional $\eta$-rule for $\Pi$-types: the rule asserting that for every \$f:($\backslash$Pi x:A)B(x), the type

\begin{displaymath}
Id(f,\lambda x,app(f,x),app(g,x))\to Id_{A\to B}(f,g)
\end{displaymath}
is inhabited.

(5) the Function extensionality axiom (FE): the rule asserting that for every $f,g:A\to B$, the type

\begin{displaymath}
(\Pi x:A) Id_B(app(f,x),app(g,x)))\to Id_{A\to B}(f,g)
\end{displaymath}
Comments:

In Voevodsky's Coq files is shown that the $\eta$-rule for dependent functions and the function extensionality principle imply the corresponding function extensionality principle for $\Sigma$-types. is inhabited.

The function extensionality axiom (FE) is implied by the univalence axiom. The system $\mathcal{H}$ does not have a global extensionality rule such as the identity reflection rule K or UIP

\hypertarget{homotopical_semantics}{}\subsection*{{Homotopical semantics}}\label{homotopical_semantics}

The \emph{transport function}: An identity term $p:Id(a,b)$ is interpreted as a path between the points $a$ and $b$. More generally an identity term $p(x):Id(a(x),b(x))$ with free variable $x$ is interpreted as a continuous family of paths, i.e. a \emph{homotopy between the continuous functions}. The identity elimination rule implies that type dependency respects identity: For a dependent type

\begin{displaymath}
x:A\vdash B(x):Type
\end{displaymath}
and an identity term $p:Id_A(a,b)$, there is a \emph{transport function}

\begin{displaymath}
p_!:B(a)\to B(b).
\end{displaymath}
For $x:A$ the function $refl(x)_!:B(x)\to B(x)$ is obtained by $Id$-elimination.

(\ldots{})

\hypertarget{extensional_types}{}\subsection*{{Extensional $W$-types}}\label{extensional_types}

The rules for $W$-types in extensional type theory are from the notes to Martin-Löf's lecture on intuitionistic type theories.

(1) $W$-formation rule

\begin{displaymath}
\frac{A:Type\; x:A\vdash B(x):Type}{(Wx:A)B(x):Type}
\end{displaymath}
In the following we will sometimes abbreviate $(Wx:A)B(x)$ by $W$

(2) $W$-introduction rule

\begin{displaymath}
\frac{a:A\; t:B(a)\to W}{sup(a,t):W}
\end{displaymath}
(3) $W$-elimination rule

\begin{displaymath}
\frac{\itexarray{w:W\vdash C(w):Type\\
x:A, u:B(x)\to W, v:(\Pi y:B(x))C(u(y))\vdash\\
c(x,y,v):C(sup(x,y))}}{w:W\vdash wrec(w,c):C(w)}
\end{displaymath}
(4) $W$-computation rule

\begin{displaymath}
\frac{\itexarray{w:W\vdash C(w):Type\\
x:A, u:B(x)\to W, v:(\Pi y:B(x))C(u(y))\vdash\\
c(x,y,v):C(sup(x,y))}}{\itexarray{
x:A, u:B(x)\to W\vdash wrec(sup(x,u),c)=\\
c(x,u,\lambda y.wrec(u(y),c)):C(sup(x,u))}}
\end{displaymath}
Comments:

We can think of $W$-types as free algebras for signatures with operations of possibly infinite arity, but no equations:

(ad1) We consider the premises of this rule as specifying a signature which has the elements of $A$ as operations and in which the arity of $a:A$ is the cardinality of the type $B(a)$.

(ad2) Then, the introduction rule -as always- specifies the canonical way of forming an element of the type in consideration.

(ad3) The elimination rule can be interpreted as the propositions-as-types translation of the appropriate induction principle.

In more detail, let $\mathcal{H}_ext$ denote the type theory obtained form $\mathcal{H}$ by adopting the identity reflection rule $\frac{p:Id_A(x,y)}{x=y:A}$ and let $\mathcal{C}(\mathcal{H}_ext)$ denote the category types as objects and function types $f:A\to B$ as morphisms where two maps are considered to be equal if they are definitionally equal.

Then, the premisses of the introduction rule determine the [[nLab:polynomial functor|polynomial endofunctor]] $P:\mathcal{C}(\mathcal{H}_ext)\to \mathcal{C}(\mathcal{H}_ext)$ defined by

\begin{displaymath}
P(X)=_{def}(\Sigma x: A)(B(x)\to X).
\end{displaymath}
A \emph{$P$-algebra} is a pair $(C,s_C)$ where $C$ is a type and $s_C:PC\to C$ is a function called the \emph{structure map of the algebra}.

The formation rule gives us an object $W=_{def} (Wx:A)B(x)$.

The introduction rule combined with the rule for $\Pi$-types and $\Sigma$-types determines a structure map $s_W:PW\to W$.

The elimination rule implies that the projection $\pi_1:C\to W$ where $C=_{def} (\Sigma w:W)C(w)$ has a section $s$ if the type $C$ has a $P$-algebra structure over $W$

The computation rule states that the section $s$ determined by the elimination rule is also a $P$-algebra homomorphism.

\begin{displaymath}
\itexarray{
PC&\stackrel{s_C}{\to}&C\\
\downarrow&&\downarrow^{\pi_1}\uparrow_s\\
PW&\stackrel{s_W}{\to}&W&
}
\end{displaymath}
The previous elimination rule implies the \emph{simple \$W-elimination rule}

\begin{displaymath}
\frac{C:Type\; x:A,v:B(x)\to C\vdash c(x,v):C}{w:W\vdash simp-wrec(w,c):C}
\end{displaymath}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Steve Awodey, Nicola gambino, Kristina Sojakova, inductive types in homotopy type theory, \href{http://arxiv.org/abs/1201.3898}{arXiv:1201.3898}


\item S. Awodey, Type theory and homotopy, \href{http://www.andrew.cmu.edu/user/awodey/preprints/TTH.pdf}{pdf}


\item T. Streicher, Investigations into intensional type theory, 1993, Habilitation Thesis. Available from the author’s web page.


\item B. Nordstrom, K. Petersson, and J. Smith, \emph{Martin-Löf type theory} in Handbook of Logic in Computer Science. Oxford Uni- versity Press, 2000, vol. 5, pp. 1–37


\item R. Garner, On the strength of dependent products in the type theory of Martin-Löf, Annals of Pure and Applied Logic, vol. 160, pp. 1–12, 2009.


\item V. Voevodsky, Univalent foundations Coq files, 2010, available from the author’s web page.


\item P. Martin-Löf, Intuitionistic Type Theory. Notes by G. Sambin of a series of lectures given in Padua, 1980. Bibliopolis, 1984.



\end{itemize}


\end{document}