\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*{directed homotopy theory}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{problem}{Problem}\dotfill \pageref*{problem} \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}

Directed Homotopy Theory is a variant of [[homotopy theory]] which aims to study the properties of [[directed space]]s. Much of the impetus for the theory comes from work on modelling concurrent process. It can also be seen as a way of studying an `evolving' space. This is discussed in more detail in the entry on [[motivation for directed homotopy]].

The following examples illustrate the sort of problems that arise:

\hypertarget{example}{}\subsection*{{Example}}\label{example}

The example uses two directed spaces that are slightly different and use a [[pospace]], i.e. a space with a closed partial order. (Both these use a rectangle with order $(x,y)\leq (x\prime, y\prime)$ if and only if $|y\prime - y| \leq x\prime - x$, so the [[future cone]] of any point is a cone symmetric about a horizontal line through the point and with edges at $\pm 45$ degrees to that line.)

The two spaces are

and

In both the space is the rectangle with two smaller rectangles removed. The position of the upper left small rectangle is the same in both, but that of the righthand lower rectangle is shifted slightly to the right in the second picture. The directed homotopy classes of d-paths from $p$ to $p^\prime$ in the two cases are different. The crucial point is that in the second there is such a class that was impossible in the first example, yet the spaces are homeomorphic, so classically would be `the same'. The subtlety is in the order.

\hypertarget{problem}{}\subsubsection*{{Problem}}\label{problem}

The first problem is to find a small model of such structures. The [[fundamental category]] would be a model, but unlike with the [[fundamental groupoid]] in the non-directed case, it is not sufficient to take a `base point' in each connected component. That would ignore the order.

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

\begin{itemize}%
\item [[directed homotopy type theory]]

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

(See also under [[directed space]].)

Foundational work was done by [[Eric Goubault]] and his collaborators.

\begin{itemize}%
\item [[Eric Goubault]], [[Some geometric perspectives in concurrency theory]].

\end{itemize}
Categorical aspects are looked at in

\begin{itemize}%
\item [[Lisbeth Fajstrup]] and [[Jiří Rosický]], \emph{A convenient category for directed homotopy}, TAC 21 no. 1, \href{http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html}{online}

\end{itemize}
For more on this see also at \emph{[[Delta-generated space]]}.

Applications to computer science are presented in

\begin{itemize}%
\item [[Lisbeth Fajstrup]], [[Eric Goubault]], [[Emmanuel Haucourt]], [[Samuel Mimram]], [[Martin Raussen]], [[Directed Algebraic Topology and Concurrency]]

\end{itemize}
The fundamental category of a pospace is discussed in

\begin{itemize}%
\item [[Lisbeth Fajstrup]], [[Eric Goubault]], [[Emmanuel Haucourt]], and [[Martin Raussen]], Components of the fundamental category. Appl.Cat. Struct. Vol. 12, pp.81-108, 2004

\end{itemize}
and the possibility of an analogue of covering spaces in

\begin{itemize}%
\item [[Lisbeth Fajstrup]], Dicovering spaces. Algebraic topological methods in computer science (Stanford, CA, 2001). Homology Homotopy Appl. 5 (2003), no. 2, 1-17 (\href{http://www.intlpress.com/HHA/v5/n2/}{HHA})

\end{itemize}
[[Philippe Gaucher]] ([[PPS]], Paris) has introduced an interesting related model, namely that of `flows'. These are, approximately, topological categories without identity arrows. They are intended as another model of processes. One of his papers on this idea is at \href{http://arxiv.org/abs/math/0308054v1}{Arxiv}, published as

\begin{itemize}%
\item [[Philippe Gaucher]], A model category for the homotopy theory of concurrency, Homology Homotopy and Applications, vol. 5 (1):p.549-599, 2003.

\end{itemize}
Marco Grandis' work on the area is listed amongst his publications at his (\href{http://www.dima.unige.it/~grandis/rec.public_grandis.html}{homepage}). Such as

\begin{itemize}%
\item [[Marco Grandis]], Directed homotopy theory. I, Cah. Topol. G eom. Di er. Cat eg. 44 (4) (2003) 281--316.


\item [[Marco Grandis]], Directed homotopy theory. II. Homotopy constructs, Theory Appl. Categ. 10 (2002) No. 14, 369--391 (electronic).


\item [[Marco Grandis]], The shape of a category up to directed homotopy, Theory Appl. Categ. 15 (2005/06) No. 4, 95--146 (electronic).


\item [[Marco Grandis]], Modelling fundamental 2-categories for directed homotopy, Homology, Homotopy Appl. 8 (1) (2006) 31--70 (electronic)


\item [[Marco Grandis]], \emph{[[Directed Algebraic Topology]], Models of non-reversible worlds} , Cambridge University Press, 2009.



\end{itemize}
A websearch will find others.

Another approach to model category structures in this area is by Kahl, who uses a Baues type fibration category approach.

\begin{itemize}%
\item Thomas Kahl, Relative directed homotopy theory of partially ordered spaces, Journal of Homotopy and Related Structures, Vol. 1(2006), No. 1, pp. 79-100,(\href{http://jhrs.rmi.acnet.ge/volumes/2006/n1a4/}{JHRS}).

\end{itemize}
[[Krzysztof Worytkiewicz]] and [[Peter Bubenik]] have given a model category structure for local pospaces:

\begin{itemize}%
\item P. Bubenik and [[K. Worytkiewicz]], A model category structure for local po-spaces, Homology, Homotopy and Applications, Vol. 8 (2006), No. 1, pp.263-292. (\href{http://intlpress.com/HHA/v8/n1/a10/}{HHA}).

\end{itemize}
Further related references are

\begin{itemize}%
\item [[L. Fajstrup]], Loops, ditopology and deadlocks, Math. Structures Comput. Sci. 10 (4) (2000) 459--480, geometry and concurrency.


\item [[L. Fajstrup]], [[M. Raussen]], [[E. Goubault]], [[E. Haucourt]], Components of the fundamental category, Appl. Categ. Structures 12 (1) (2004) 81--108, homotopy theory.


\item [[L. Fajstrup]], Dihomotopy classes of dipaths in the geometric realization of a cubical set: from discrete to continuous and back again, in: R. Kopperman, M. B. Smyth, D. Spreen, J. Webster (Eds.), \emph{Spatial Representation: Discrete vs. Continuous Computational Models}, no. 04351 in Dagstuhl Seminar Proceedings, IBFI, Schloss Dagstuhl, Germany, 2005.


\item [[L. Fajstrup]], Dipaths and dihomotopies in a cubical complex, Adv. in Appl. Math. 35 (2) (2005) 188--206.


\item [[M. Raussen]], Deadlocks and dihomotopy in mutual exclusion models, in: R. Kopperman, M. B. Smyth, D. Spreen, J. Webster (Eds.), \emph{Spatial Representation: Discrete vs. Continuous Computational Models, no. 04351 in Dagstuhl Seminar Proceedings}, IBFI, Schloss Dagstuhl, Germany, 2005.


\item U. Fahrenberg, [[M. Raussen]], Reparametrizations of continuous paths, available as \href{http://www.math.aau.dk/research/reports/2006.htm}{preprint R-2006-22} (2006).


\item [[E. Goubault]], [[E. Haucourt]], Directed algebraic topology and concurrency, (\href{http://iml.univ-mrs.fr/lafont/Geocal/goubault2.pdf}{web}) (2006).23


\item [[E. Goubault]], [[Emmanuel Haucourt|E. Haucourt]], Components of the fundamental category, II, technical reports, CEA, Saclay (2006).


\item [[M. Raussen]], Invariants of directed spaces, available as preprint R-2006-28 (\href{http://bib.mathematics.dk/preprint.php?id=DMF-2006-08-002}{web})(2006).



\end{itemize}
[[!redirects directed algebraic topology]]



\end{document}