\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{bracket 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{in_type_theory}{In type theory}\dotfill \pageref*{in_type_theory} \linebreak
\noindent\hyperlink{DefinitionInHomotopyTypeTheory}{In homotopy type theory}\dotfill \pageref*{DefinitionInHomotopyTypeTheory} \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}

There are various different paradigms for the interpretation of [[predicate logic]] in [[type theory]]. In ``logic-enriched type theory'', there is a separate class of ``[[propositions]]'' from the class of ``[[types]]''. But we can also identify propositions with particular types. In the \emph{[[propositions as types]]}-paradigm, every proposition is a type, and also every type is identified with a proposition (the proposition that it is an [[inhabited type]]).

By contrast, in the paradigm that may be called \href{propositions+as+types#PropositionsAsSomeTypes}{propositions as some types}, every proposition is a type, but not every type is a proposition. The types which are propositions are generally those which ``have at most one inhabitant'' --- in [[homotopy type theory]] this is called being of [[h-level 1]] or being a [[homotopy n-type|(-1)-type]]. This paradigm is often used in the [[categorical semantics]] of type theory, such as the [[internal logic]] of various kinds of categories.

Under ``propositions as types'', all type-theoretic operations represent corresponding logical operations ([[dependent sum]] is the [[existential quantifier]], [[dependent product]] the [[universal quantifier]], and so on). However, under ``propositions as some types'', not every such operation preserves the class of propositions; this is particularly the case for [[dependent sum]] and [[disjunction]]([[or]]). Thus, in order to obtain the correct logical operations, we need to reflect these constructions back into propositions somehow, finding the ``underlying proposition'', corresponding to the [[truncated|(-1)-truncation]]/[[h-level 1|h-level 1-projection]]. This operation in type theory is called the \textbf{bracket type} (when denoted $[A]$); in [[homotopy type theory]] it can be identified with the [[higher inductive type]] $isInhab$.

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

\hypertarget{in_type_theory}{}\subsubsection*{{In type theory}}\label{in_type_theory}

(\ldots{})

\hypertarget{DefinitionInHomotopyTypeTheory}{}\subsubsection*{{In homotopy type theory}}\label{DefinitionInHomotopyTypeTheory}

We discuss the definition in [[homotopy type theory]].

For $A$ a type, the \textbf{support} of $A$ denoted $supp(A)$ or $isInhab(A)$ or $\tau_{-1} A$ or $\| A \|_{-1}$ or $\| A \|$ or, lastly, $[A]$, is the [[higher inductive type]] defined by the two constructors

\begin{displaymath}
a \colon A \;\vdash \; isinhab(a) \colon supp(A)
\end{displaymath}
\begin{displaymath}
x \colon supp(A) \;,\; y \colon supp(A)
  \;\vdash \;
  inpath(x,y) \colon (x = y)
 \,,
\end{displaymath}
where in the last [[sequent]] on the right we have the [[identity type]]. (\hyperlink{Voevodsky}{Voevodsky}, \hyperlink{HoTTLibrary}{HoTTLibrary})

This says that $supp(A)$ is the type which is [[universal property|universal]] with the property that the [[terms]] of $A$ map to it and that any two term of $A$ become [[equivalence in homotopy type theory|equivalent]] in $supp(A)$.

In [[Agda]] [[syntax]] this is

\begin{verbatim}data isinhab {i : Level} (A : Set i) : Set i where
  inhab : A → isinhab A
  inhab-path : (x y : isinhab A) → x ≡ y\end{verbatim}
The [[recursion principle]] for $supp(A)$ says that if $B$ is a [[mere proposition]] and we have $f: A \to B$, then there is an induced $g : supp(A) \to B$ such that $g(isinhab(a)) \equiv f(a)$ for all $a:A$. In other words, any mere proposition which follows from (the inhabitedness of) $A$ already follows from $supp(A)$. Thus, $supp(A)$, as a mere proposition, contains no more information than the inhabitedness of $A$.

For more see at \emph{[[n-truncation modality]]}.

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

One presentation of the [[internal logic|internal]] [[type theory]] of [[regular categories]] consists of [[dependent type theory]] with the [[unit type]], [[strong extensional equality types]], strong [[dependent sums]], and bracket types. (The internal logic of a regular category can alternatively be presented as a [[logic-enriched type theory]].)

The [[semantics]] of bracket types in a [[regular category]] $C$ is as follows.

A [[dependent type]] (a type in [[context]] $X$)

\begin{displaymath}
x\colon X \vdash A(x) \colon Type
\end{displaymath}
is interpreted in $C$ as an arbitrary [[morphism]]

\begin{displaymath}
\itexarray{
    A
    \\  
    \downarrow
    \\
    X
  }
  \,.
\end{displaymath}
The corresponding bracket type

\begin{displaymath}
x\colon X \vdash [A(x)] \colon Type
\end{displaymath}
is interpreted then as the [[image]]-[[(epi, mono) factorization system|factorization]]

\begin{displaymath}
\itexarray{
     A &&\to&& [A] & := im(A \to X)
     \\
     & \searrow && \swarrow
     \\
     && X
    \,.
  }
\end{displaymath}
Therefore $[A] \to X$ is a [[monomorphism]], and hence the interpretation of a [[proposition]] about the elements of $X$.

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

\begin{itemize}%
\item [[image]], [[inhabited type]]


\item [[n-truncation modality]]



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

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

The original articles are

\begin{itemize}%
\item M.E. Maietti, \emph{The Type Theory of Categorical Universes} PhD thesis, Universit\`a{} Delgi Studi di Padova, 1998

\end{itemize}
(which speaks of ``mono types'') and

\begin{itemize}%
\item [[Frank Pfenning]], \emph{Intensionality, extensionality, and proof irrelevance in modal type theory}, In Proceedings of the 16th Annual Symposium on Logic in Computer Science (LICS'01), June 2001.


\item [[Steve Awodey]], [[Andrej Bauer]], \emph{Propositions as $[$Types$]$}, Journal of Logic and Computation. Volume 14, Issue 4, August 2004, pp. 447-471 (\href{http://andrej.com/papers/brackets_letter.pdf}{pdf})



\end{itemize}
Exposition in the context of [[homotopy type theory]] is in

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

\end{itemize}
Formalization in the context of homotopy type theory is in

\begin{itemize}%
\item [[Vladimir Voevodsky]], \emph{The hProp version of the ``inhabited'' construction.} (\href{http://www.math.ias.edu/~vladimir/Foundations_library/hProp.html#lab140}{web})

\end{itemize}
\begin{itemize}%
\item [[Coq]] \href{https://github.com/HoTT/HoTT/tree/master/Coq}{HoTT library}, \emph{\href{https://github.com/HoTT/HoTT/blob/master/Coq/HIT/IsInhab.v}{IsInhab.v}}

\end{itemize}
Discussion of this in the more general context of truncations is in

\begin{itemize}%
\item [[Guillaume Brunerie]], \emph{Truncations and truncated higher inductive types} (\href{http://homotopytypetheory.org/2012/09/16/truncations-and-truncated-higher-inductive-types/}{web})

\end{itemize}
[[!redirects bracket types]] [[!redirects propositions as some types]]

[[!redirects propositional truncation]] [[!redirects propositional truncations]] [[!redirects squash type]] [[!redirects squash types]] [[!redirects isInhab]]



\end{document}