PlanetPhysics/Cubically Thin Homotopy 2

Cubically thin homotopyEdit

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

    i.e. Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',}

  • is rel. vertices of

    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,

  • 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 SourcesEdit



  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.