cubically thin homotopy

0.1 Cubically thin homotopy

Let u,u be squares in X with common vertices.

  1. 1.

    A cubically thin homotopy U:uTu between u and u is a cube ( UR3(X) such that

    • U is a homotopyMathworldPlanetmathPlanetmath between u and u,

      i.e. 1-(U)=u,1+(U)=u,

    • U is rel. vertices of I2,

      i.e. 2-2-(U),2-2+(U),2+2-(U),2+2+(U) are constant,

    • the faces iα(U) are thin for α=±1,i=1,2.

  2. 2.

    The square u is cubically T-equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to u, denoted uTu if there is a cubically thin homotopy between u and u.

This definition enables one to construct the homotopy double groupoidPlanetmathPlanetmath scheme 𝝆2(X) , by defining a relationMathworldPlanetmathPlanetmathPlanetmath of cubically thin homotopy on the set R2(X) of squares.


  • 1 K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, Applied Cat. StructuresMathworldPlanetmath, 8 (2000): 209-234.
  • 2 R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, Theory and Applications of Categories 10,(2002): 71-93.
Title cubically thin homotopy
Canonical name CubicallyThinHomotopy
Date of creation 2013-03-22 18:15:06
Last modified on 2013-03-22 18:15:06
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 17
Author bci1 (20947)
Entry type Definition
Classification msc 55N33
Classification msc 55N20
Classification msc 55U40
Classification msc 18D05
Synonym higher dimensional thin homotopy
Related topic HomotopyDoubleGroupoidOfAHausdorffSpace
Related topic HomotopyAdditionLemma
Related topic WeakHomotopyAdditionLemma
Related topic Polyhedron
Defines higher dimensional thin Homotopy