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\begin{document}

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\section*{unit type}

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

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

[[!include type theory - 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{as_a_positive_type}{As a positive type}\dotfill \pageref*{as_a_positive_type} \linebreak
\noindent\hyperlink{as_a_negative_type}{As a negative type}\dotfill \pageref*{as_a_negative_type} \linebreak
\noindent\hyperlink{positive_versus_negative}{Positive versus negative}\dotfill \pageref*{positive_versus_negative} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{categorical_semantics}{Categorical semantics}\dotfill \pageref*{categorical_semantics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

In [[type theory]] the \emph{unit type} is the [[type]] with a unique [[term]]. It is the special case of a [[product type]] with no factors.

In a [[model]] by [[categorical semantics]], this is a [[terminal object]]. In [[set theory]], it is a [[singleton]].

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

Like any type in type theory, the unit type is specified by rules saying when we can introduce it as a type, how to construct terms of that type, how to use or ``eliminate'' terms of that type, and how to compute when we combine the constructors with the eliminators.

The unit type, like the binary [[product type]], can be presented both as a [[positive type]] and a [[negative type]]. In both cases the rule for building the unit type is the same, namely ``it exists'':

\begin{displaymath}
\frac{ }{1\colon Type}
\end{displaymath}
\hypertarget{as_a_positive_type}{}\subsubsection*{{As a positive type}}\label{as_a_positive_type}

Regarded as a positive type, we give primacy to the constructors, of which there is exactly one, denoted $()$ or $tt$.

\begin{displaymath}
\frac{ }{() \colon 1 }
\end{displaymath}
The eliminator now says that to use something of type $1$, it suffices to say what to do in the case when that thing is $()$.

\begin{displaymath}
\frac{\vdash c\colon C}{u\colon 1 \vdash let () = u in c \;\colon C}
\end{displaymath}
Obviously this is not very interesting, but this is what we get from the general rules of type theory in this degenerate case. In [[dependent type theory]], we should also allow $C$ to depend on the unit type $1$.

We then have the [[∞-reduction]] rule:

\begin{displaymath}
let () = () in c \;\to_\beta\; c
\end{displaymath}
and (if we wish) the [[∞-reduction]] rule:

\begin{displaymath}
let () = u in c[()/z] \;\to_\eta\; c[u/z].
\end{displaymath}
The $\beta$-reduction rule is simple. The $\eta$-reduction rule says that if $c$ is an expression involving a variable $z$ of type $1$, and we substitute $()$ for $z$ in $c$ and use that (with the eliminator) to define a term involving another term $u$ of type $1$, then the result is the same as what we would get by substituting $u$ for $z$ in $c$ directly.

The positive presentation of the unit type is naturally expressed as an [[inductive type]]. In [[Coq]] syntax:

\begin{verbatim}Inductive unit : Type :=
| tt : unit.\end{verbatim}
(Coq then implements beta-reduction, but not eta-reduction. However, eta-equivalence is provable with the internally defined [[identity type]], using the dependent eliminator mentioned above.)

\hypertarget{as_a_negative_type}{}\subsubsection*{{As a negative type}}\label{as_a_negative_type}

A negative type is characterized by its eliminators, which is a little subtle for the unit type. But by analogy with binary [[product types]], which have two eliminators $\pi_1$ and $\pi_2$ when presented negatively, the negative unit type (a [[zero|nullary]] product) should have \emph{no eliminators at all}.

To derive the constructors from this, we follow the general rule for negative types that to construct an element of $1$, it should suffice to specify how that element behaves under all the eliminators. Since there \emph{are} no eliminators, this means we have nothing to do; thus we have exactly one way to construct an element of $1$, by doing nothing:

\begin{displaymath}
\frac{ }{()\colon 1}
\end{displaymath}
There is no $\beta$-reduction rule, since there are no eliminators to compose with the constructor. However, there \emph{is} an $\eta$-conversion rule. In general, an $\eta$-[[redex]] consists of a constructor whose inputs are all obtained from eliminators. Here we have a constructor which has no inputs, so it is not a problem that we have no eliminators to fill them in with. Thus, the term $()\colon 1$ is an ``$\eta$-redex'', and it ``reduces'' to \emph{any} term of type $1$ at all:

\begin{displaymath}
() \;\to_\eta\; u
\end{displaymath}
This is obviously not a well-defined ``operation'', so in this case it is better to view $\eta$-conversion in terms of \emph{expansion}:

\begin{displaymath}
u \;\to_\eta\; ().
\end{displaymath}
\hypertarget{positive_versus_negative}{}\subsubsection*{{Positive versus negative}}\label{positive_versus_negative}

In ordinary ``nonlinear'' type theory, the positive and negative unit types are equivalent. They manifestly have the same constructor, while we can define the positive eliminator in a trivial way as

\begin{displaymath}
let () = u in c  \quad \coloneqq\quad c
\end{displaymath}
Note that just as for binary [[product types]], in order for this to be a well-typed definition of the \emph{dependent} eliminator in dependent type theory, we need to assume the $\eta$-conversion rule for the assumed negative unit type (at least, \href{/nlab/show/eta-conversion#Propositional}{propositionally}).

Of course, the positive $\beta$-reduction rule holds by definition for the above defined eliminator.

For the $\eta$-conversion rules, if we start from the negative presentation and define the positive eliminator as above, then $let () = u in c[()/z]$ is precisely $c[()/z]$, which is convertible to $c[u/z]$ by the negative $\eta$-conversion rule $() \;\leftrightarrow_\eta\; u$.

Conversely, if we start from the positive presentation, then we have

\begin{displaymath}
()
  \;\leftrightarrow_\eta\; let () = u in ()
  \;\leftrightarrow_\eta\; u
\end{displaymath}
where in the first conversion we use $c \coloneqq ()$ and in the second we use $c\coloneqq z$.

As in the case of binary [[product types]], these translations require the [[contraction rule]] and the [[weakening rule]]; that is, they duplicate and discard terms. In [[linear logic]] these rules are disallowed, and therefore the positive and negative unit types become different. The positive product becomes ``one'' $\mathbf{1}$, while the negative product becomes ``top'' $\top$.

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

\begin{prop}
\label{}\hypertarget{}{}
In [[homotopy type theory]] the unit type is a [[contractible type]], and every contractible type is [[equivalence in homotopy type theory|equivalent]] to the unit type.

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

Under [[categorical semantics]], a unit type satisfying both beta and eta conversions corresponds to a [[terminal object]] in a [[category]]. More precisely:

\begin{itemize}%
\item A terminal object may be used to interpret a unit type that validates both beta and eta rules, while


\item the syntactic category of a type theory with a unit type has a terminal object, as long as the unit type satisfies both beta and eta rules.



\end{itemize}
Of course, the categorical notion of terminal object matches the \emph{negative} definition of a unit type most directly. In [[linear logic]], therefore, the categorical terminal object interprets ``top'' $\top$, while the unit object of an additional [[monoidal category|monoidal structure]] interprets ``one'' $\mathbf{1}$.

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

\begin{itemize}%
\item [[product type]]
\item [[empty type]]
\item [[contractible type]]

\end{itemize}
[[!include homotopy n-types - table]]

[[!redirects unit type]] [[!redirects unit types]] [[!redirects trivial type]] [[!redirects trivial types]]



\end{document}