Talk:PlanetPhysics/Homotopy Category

Original TeX Content from PlanetPhysics Archive

edit
%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: homotopy category
%%% Primary Category Code: 00.
%%% Filename: HomotopyCategory.tex
%%% Version: 1
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetPhysics preamble. as your 
% almost certainly you want these

\usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate}
\usepackage{xypic, xspace}
\usepackage[mathscr]{eucal}
\usepackage[dvips]{graphicx}
\usepackage[curve]{xy}

% define commands here
\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
%\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem*{notation}{Notation}
\newtheorem*{claim}{Claim}
\renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}}
\numberwithin{equation}{section}
\newcommand{\Ad}{{\rm Ad}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Cl}{{\rm Cl}}
\newcommand{\Co}{{\rm Co}}
\newcommand{\DES}{{\rm DES}}
\newcommand{\Diff}{{\rm Diff}}
\newcommand{\Dom}{{\rm Dom}}
\newcommand{\Hol}{{\rm Hol}}
\newcommand{\Mon}{{\rm Mon}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Ind}{{\rm Ind}}
\newcommand{\IM}{{\rm Im}}
\newcommand{\Is}{{\rm Is}}
\newcommand{\ID}{{\rm id}}
\newcommand{\grpL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\rO}{{\rm O}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\SL}{{\rm Sl}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\Symb}{{\rm Symb}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}
\newcommand{\A}{\mathcal A}
\newcommand{\Ce}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
%\newcommand{\grp}{\mathcal G}
\renewcommand{\H}{\mathcal H}
\renewcommand{\cL}{\mathcal L}
\newcommand{\Q}{\mathcal Q}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}

\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bO}{\mathbb{O}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bV}{\mathbb{V}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bfE}{\mathbf{E}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfY}{\mathbf{Y}}
\newcommand{\bfZ}{\mathbf{Z}}
\renewcommand{\O}{\Omega}
\renewcommand{\o}{\omega}
\newcommand{\vp}{\varphi}
\newcommand{\vep}{\varepsilon}
\newcommand{\diag}{{\rm diag}}
\newcommand{\grp}{{\mathsf{G}}}
\newcommand{\dgrp}{{\mathsf{D}}}
\newcommand{\desp}{{\mathsf{D}^{\rm{es}}}}
\newcommand{\grpeod}{{\rm Geod}}
%\newcommand{\grpeod}{{\rm geod}}
\newcommand{\hgr}{{\mathsf{H}}}
\newcommand{\mgr}{{\mathsf{M}}}
\newcommand{\ob}{{\rm Ob}}
\newcommand{\obg}{{\rm Ob(\mathsf{G)}}}
\newcommand{\obgp}{{\rm Ob(\mathsf{G}')}}
\newcommand{\obh}{{\rm Ob(\mathsf{H})}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\grphomotop}{{\rho_2^{\square}}}
\newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}}
\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\grplob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\grpa}{\grpamma}
%\newcommand{\grpa}{\grpamma}
\newcommand{\de}{\delta}
\newcommand{\del}{\partial}
\newcommand{\ka}{\kappa}
\newcommand{\si}{\sigma}
\newcommand{\ta}{\tau}
\newcommand{\lra}{{\longrightarrow}}
\newcommand{\ra}{{\rightarrow}}
\newcommand{\rat}{{\rightarrowtail}}
\newcommand{\ovset}[1]{\overset {#1}{\ra}}
\newcommand{\ovsetl}[1]{\overset {#1}{\lra}}
\newcommand{\hr}{{\hookrightarrow}}
\newcommand{\<}{{\langle}}

%\usepackage{geometry, amsmath,amssymb,latexsym,enumerate}
%\usepackage{xypic}

\def\baselinestretch{1.1}


\hyphenation{prod-ucts}

%\grpeometry{textwidth= 16 cm, textheight=21 cm}

\newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}&
#3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram
\eqno{\mbox{#9}}$$ }

\def\C{C^{\ast}}

\newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}}

%\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$
%{\mbox{}}

\newcommand{\quadr}[4]
{\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4&
\end{pmatrix}}
\def\D{\mathsf{D}}

\begin{document}

 \subsection{Homotopy category, fundamental groups and fundamental groupoids}

Let us consider first the \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} \textbf{$Top$} whose \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} spaces $X$ with a chosen basepoint $x \in X$ and whose \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are continuous maps $X \to Y$ that associate the basepoint of $Y$ to the
basepoint of $X$. The fundamental group of $X$ specifies a \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $Top \to \textbf{G}$, with $\textbf{G}$ being the category of \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and group \htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, which is called \emph{the fundamental group functor}.

\subsection{Homotopy category}
Next, when one has a suitably defined \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} between morphisms, or maps, in a category \textbf{$U$}, one can define the \emph{homotopy category} $hU$ as the category whose objects are the same as the objects of \textbf{$U$}, but with morphisms being defined by the homotopy classes of maps; this is in fact the homotopy category of \emph{unbased spaces}.

\subsection{Fundamental groups}
We can further require that homotopies on \textbf{$Top$} map each basepoint to a corresponding basepoint, thus leading to the definition of the \emph{homotopy category $hTop$ of based spaces}. Therefore, the fundamental group is a \emph{homotopy invariant} functor on \textbf{$Top$}, with the meaning that the latter functor factors through a functor $ hTop \to \textbf{G} $. A homotopy equivalence in \textbf{$U$} is an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} in $hTop$. Thus, based homotopy equivalence induces an isomorphism of fundamental groups.

\subsection{Fundamental groupoid}
In the general case when one does not choose a basepoint, a \emph{\htmladdnormallink{fundamental groupoid}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}} $\Pi_1 (X)$ of a topological space $X$ needs to be defined as the category whose objects are the base points of $X$ and whose morphisms $x \to y$ are the equivalence classes of paths from $x$ to $y$.

\begin{itemize}
\item Explicitly, the objects of $\Pi_1(X)$ are the points of $X$
$$\mathrm{Obj}(\Pi_1(X))=X\,,$$
\item morphisms are homotopy classes of paths ``rel endpoints'' that is
$$\mathrm{Hom}_{\Pi_1(x)}(x,y)=\mathrm{Paths}(x,y)/\sim\, ,$$
where, $\sim$ denotes homotopy rel endpoints, and,
\item \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} of morphisms is defined \emph{via} piecing together, or concatenation, of paths.
\end{itemize}

\subsection{Fundamental groupoid functor}

Therefore, the set of endomorphisms of an object $x$ is precisely the fundamental group $\pi(X,x)$. One can thus construct the \emph{\htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} of homotopy equivalence classes}; this construction can be then carried out by utilizing functors from the category \textbf{$Top$}, or its subcategory $hU$,
to the \emph{\htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory4.html} and \htmladdnormallink{groupoid homomorphisms}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}}, $Grpd$. One such functor
which associates to each topological space its fundamental (homotopy) groupoid is appropriately called the
\emph{\htmladdnormallink{fundamental groupoid functor}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid.html}}.

\subsection{An example: the category of simplicial, or CW-complexes}

As an important example, one may wish to consider the category of \htmladdnormallink{simplicial}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html}, or $CW$-complexes and homotopy defined
for $CW$-complexes. Perhaps, the simplest example is that of a one-dimensional $CW$-complex, which is a \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Cod.html}.
As described above, one can define a functor from the category of graphs, \textbf{Grph}, to \textbf{$Grpd$} and then define the fundamental homotopy groupoids of graphs, \htmladdnormallink{hypergraphs}{http://planetphysics.us/encyclopedia/SimpleIncidenceStructure2.html}, or pseudographs. The case of freely generated graphs (one-dimensional $CW$-complexes) is particularly simple and can be computed with a digital \htmladdnormallink{computer}{http://planetphysics.us/encyclopedia/Program3.html} by a finite \htmladdnormallink{algorithm}{http://planetphysics.us/encyclopedia/RecursiveFunction.html} using the finite groupoids associated with such finitely generated $CW$-complexes.

\subsubsection{Remark}
Related to this \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of homotopy category for unbased topological spaces, one can then prove the \emph{\htmladdnormallink{approximation theorem for an arbitrary space}{http://planetphysics.us/encyclopedia/ApproximationTheoremForAnArbitrarySpace.html}} by considering a functor $$\Gamma : \textbf{hU} \longrightarrow \textbf{hU},$$ and also the construction of an approximation of an arbitrary space $X$ as the
colimit $\Gamma X$ of a sequence of cellular inclusions of $CW$-complexes $X_1, ..., X_n$ , so that one obtains $X \equiv colim [X_i]$.

Furthermore, the \htmladdnormallink{homotopy groups}{http://planetphysics.us/encyclopedia/ExtendedHurewiczFundamentalTheorem.html} of the $CW$-complex $\Gamma X$ are the colimits of the homotopy groups of $X_n$, and $\gamma_{n+1} : \pi_q(X_{n+1})\longmapsto\pi_q (X)$ is a group \htmladdnormallink{epimorphism}{http://planetphysics.us/encyclopedia/Epimorphism2.html}.

\begin{thebibliography}{9}

\bibitem{MJP1999}
May, J.P. 1999, \emph{A Concise Course in Algebraic Topology.}, The University of Chicago Press: Chicago

\bibitem{BR-JG2k4}
R. Brown and G. Janelidze.(2004). Galois theory and a new homotopy double groupoid of a map of spaces.(2004).
{\em Applied Categorical Structures},\textbf{12}: 63-80. Pdf file in arxiv: math.AT/0208211

\end{thebibliography} 

\end{document}
Return to "PlanetPhysics/Homotopy Category" page.