\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*{split support}

\hypertarget{split_supports}{}\section*{{Split supports}}\label{split_supports}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{relation_to_axiom_of_choice}{Relation to axiom of choice}\dotfill \pageref*{relation_to_axiom_of_choice} \linebreak
\noindent\hyperlink{in_foundations}{In foundations}\dotfill \pageref*{in_foundations} \linebreak
\noindent\hyperlink{in_set_theory}{In set theory}\dotfill \pageref*{in_set_theory} \linebreak
\noindent\hyperlink{in_homotopy_type_theory}{In (homotopy) type theory}\dotfill \pageref*{in_homotopy_type_theory} \linebreak
\noindent\hyperlink{objects_with_split_support}{Objects with split support}\dotfill \pageref*{objects_with_split_support} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Recall that the \textbf{[[support]]} of an object $X$ of a [[regular category]] (or [[n-category]]) is its [[(-1)-truncation]], i.e. the [[image]] of the map $X\to 1$ to the [[terminal object]]. Let $\Vert X \Vert$ denote its support.

We say that $X$ has a \textbf{split support} if the canonical map $X\to {\Vert X \Vert}$ has a [[section]]. Since $\Vert X \Vert$ is a [[subterminal object]], it is equivalent to say that there exists any morphism ${\Vert X \Vert} \to X$.

If all objects has split supports, we say that \textbf{supports split} in the ambient category.

\hypertarget{relation_to_axiom_of_choice}{}\subsection*{{Relation to axiom of choice}}\label{relation_to_axiom_of_choice}

Supports splitting in a category is a weak form of the external [[axiom of choice]] (all regular epis split). In fact, the splitting of supports is exactly the ``difference'' between the external axiom of choice and the \emph{internal} axiom of choice, i.e.

\begin{displaymath}
EAC \Leftrightarrow (IAC \;\text{ and }\; SS).
\end{displaymath}
\hypertarget{in_foundations}{}\subsection*{{In foundations}}\label{in_foundations}

\hypertarget{in_set_theory}{}\subsubsection*{{In set theory}}\label{in_set_theory}

If we assume [[excluded middle]], then supports split in the category [[Set]]. This has little to do with the foundational [[axiom of choice]]; it is more like [[well-pointed category|well-pointedness]]. Note, however, that to say there is a \emph{function} assigning to every $X$ a section of $X\to {\Vert X \Vert}$ is much stronger: such a function is a global [[choice operator]].

In [[constructive mathematics]], however, it may not be true that all supports split in Set; this can fail in the [[internal logic]] of some [[toposes]]. (Note that the ``internal'' version of ``supports split in Set'' in a topos may not be the same as the statement that supports split as an external statement about the topos itself.) See \hyperlink{FourmanScedrov}{(Fourman-Scedrov)} and \hyperlink{KECA}{(KECA)}. Thus, splitting of supports can be regarded as a weaker form of [[excluded middle]].

\hypertarget{in_homotopy_type_theory}{}\subsubsection*{{In (homotopy) type theory}}\label{in_homotopy_type_theory}

In [[type theory]] under [[propositions as types]]), where assertions of existence are always witnessed, to say that ``all supports split'' would by default mean that there is a \emph{function} as above, assigning to every $X$ a section of $X\to {\Vert X \Vert}$, and imply the global axiom of choice. To recover the statement which depends only on excluded middle, we need an additional truncation:

\begin{displaymath}
\prod_{(X:Type)} \Vert ( \Vert X \Vert \to X ) \Vert.
\end{displaymath}
We might pronounce this version as ``all supports [[mere proposition|merely]] split''.

In [[homotopy type theory]], the pure constructive version of ``all supports split'' ($\prod_{(X:Type)} \Vert X \Vert \to X$) is in fact inconsistent: it contradicts the [[univalence axiom]]. As before, the truncated version is true under LEM but may fail otherwise.

\hypertarget{objects_with_split_support}{}\paragraph*{{Objects with split support}}\label{objects_with_split_support}

Since not all supports split in homotopy type theory, it is interesting to consider whether the support of some particular type is split. For instance, for any $f:A\to B$, the type $qinv(f) \coloneqq \sum_{g:B\to A} (f g = id_B) \times (g f = id_A)$ has split support, since its support can be proven to be equal to the type of coherent equivalence data on $f$.

It is shown in \hyperlink{KECA}{(KECA)} that a type in homotopy type theory has split support if and only if it admits a [[steady function|steady]] endomap. Thus, it has merely split support if and only if it merely admits a steady endomap.

Some general classes of types can be shown \emph{not} to have split support, at least not uniformly, by arguments similar to the one which shows that not all types have split support. For instance, the identity type $x=_A y$ has split support for all $x,y:A$ if and only if $A$ is an [[h-set]]; this is proven in \hyperlink{KECA}{(KECA)}, and the ``only if'' direction is also a special case of Theorem 7.2.2 of [[the HoTT book]].

Similarly, we have:

\begin{utheorem}
If the type $steady(f)\coloneqq \prod_{(x,y:A+A)} (f x = f y)$ has split support for every [[endomap]] $f:A+A\to A+A$, then $A$ is an [[h-set]].

\end{utheorem}
\begin{proof}
Given $a,b:A$ with $\Vert a=b\Vert$, we show $a=b$; this suffices to show that $A$ is a set. Define $g,h : A \to A+A$ to be constant at $inl(a)$ and $inl(b)$ respectively, and $f = [g,h] : A+A \to A+A$. Then

\begin{displaymath}
steady(f) =  ( (A\times A) \to ((a=a) \times (a=b) \times (b=a) \times (b=b)) )
\end{displaymath}
Since $\Vert a=b\Vert$, we have $\Vert steady(f)\Vert$. Thus, by assumption, $steady(f)$; hence $a=b$.

\end{proof}
In particular, not all types of the form $steady(f)$ have split support. Thus, ``steadiness'' is not as well-behaved a notion as being quasi-invertible.

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

\begin{itemize}%
\item M. P. Fourman and A. Scedrov. The ``world's simplest axiom of choice'' fails. \emph{Manuscripta Math.}, 38(3):325\{332, 1982.


\item [[Nicolai Kraus]], [[Martin Escardo]], [[Thierry Coquand]], [[Thorsten Altenkirch]], \emph{Generalizations of Hedberg's theorem}, in M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. \href{http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf}{PDF}.

(In this paper, types with split support are called ``h-stable'')



\end{itemize}
[[!redirects split support]] [[!redirects split supports]] [[!redirects supports split]]

[[!redirects h-stable type]] [[!redirects h-stable types]]



\end{document}