Talk:PlanetPhysics/Homotopy Category

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\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} 

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