cubically thin homotopy

0.1 Cubically thin homotopy

Let $u,u^{\prime}$ be squares in $X$ with common vertices.

1.

A cubically thin homotopy $U:u\equiv^{\square}_{T}u^{\prime}$ between $u$ and $u^{\prime}$ is a cube (http://planetmath.org/Polyhedron) $U\in R^{\square}_{3}(X)$ such that
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  $U$ is a homotopy between $u$ and $u^{\prime},$
  
  i.e. $\partial^{-}_{1}(U)=u,\enskip\partial^{+}_{1}(U)=u^{\prime},$
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  $U$ is rel. vertices of $I^{2},$
  
  i.e. $\partial^{-}_{2}\partial^{-}_{2}(U),\enskip\partial^{-}_{2}\partial^{+}_{2}(U)% ,\enskip\partial^{+}_{2}\partial^{-}_{2}(U),\enskip\partial^{+}_{2}\partial^{+% }_{2}(U)$ are constant,
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  the faces $\partial^{\alpha}_{i}(U)$ are thin for $\alpha=\pm 1,\ i=1,2$ .
2.

The square $u$ is cubically $T$ -equivalent to $u^{\prime},$ denoted $u\equiv^{\square}_{T}u^{\prime}$ if there is a cubically thin homotopy between $u$ and $u^{\prime}.$

This definition enables one to construct the homotopy double groupoid scheme $\boldsymbol{\rho}^{\square}_{2}(X)$ , by defining a relation of cubically thin homotopy on the set $R^{\square}_{2}(X)$ of squares.

References

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, 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