weak homotopy double groupoidWeakHomotopyDoubleGroupoid2008-07-20 14:51:072008-10-18 15:33:28Definitionhigher dimensional weak homotopyhigher dimensional weak homotopyhigher dimensional algebraHDAweak homotopy double groupoid% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\hr}{{\hookrightarrow}}\begin{definition}
\emph{a weak homotopy double groupoid (WHDG)} of a
\emph{compactly--generated space} $X _{cg}$, (weak Hausdorff space) is
defined through a construction method similar to that developed by R. Brown (ref. \cite{BHKP}) for the \emph{homotopy double groupoid of a Hausdorff space}. The key changes here involve replacing the regular homotopy equivalence relation from the cited ref. with the \emph{weak homotopy equivalence relation} in the definition of the fundamental groupoid, as well as replacing the Hausdorff space by the compactly-generated space $X_{cg}$. Therefore, the weak homotopy data for the \emph{weak homotopy double groupoid} of $X_{cg}$, $\boldsymbol{\rho}^{\square} (X_{cg})$, will now be: \\
\end{definition}
\[
\begin{array}{c}
(\boldsymbol{\rho}^{\square}_2 (X), \boldsymbol{\rho}_1^{\square} (X) ,
\partial^{-}_{1} , \partial^{+}_{1} , +_{1} , \varepsilon _{1}) ,
\boldsymbol{\rho}^{\square}_2 (X), \boldsymbol{\rho}^{\square}_1 (X) ,
\partial^{-}_{2} , \partial^{+}_{2} , +_{2} , \varepsilon _{2})\\[3mm]
(\boldsymbol{\rho}^{\square}_1 (X) , X , \partial^{-} , \partial^{+} , + , \varepsilon).
\end{array}\]
\begin{thebibliography}{9}
\bibitem{BHKP}
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff
space, {\it Theory and Applications of Categories} \textbf{10},(2002): 71-93.
\end{thebibliography}