PlanetPhysics/Cubically Thin Homotopy 2

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Cubically thin homotopy[edit]

Let be squares in with common vertices.


  1. A {\it cubically thin homotopy}

between and is a cube such that

 #
  • is a homotopy between and \begin{center} i.e. Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',} \end{center} #
  • is rel. vertices of \begin{center} i.e. Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_2\partial^{-}_2 (U),\enskip\partial^{-}_2 \partial^{+}_2 (U),\enskip \partial^{+}_2\partial^{-}_2 (U),\enskip\partial^{+}_2 \partial^{+}_2 (U)} are constant,\end{center} #
  • the faces are thin for .
  1. The square is {\it cubically} -{\it equivalent} to

denoted if there is a cubically thin homotopy between and


This definition enables one to construct , by defining a relation of cubically thin homotopy on the set of squares.

All Sources[edit]

[1] [2]

References[edit]

  1. K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, Applied Cat. Structures , 8 (2000): 209-234.
  2. 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} 10 ,(2002): 71-93.