cubically thin homotopy
0.1 Cubically thin homotopy
Let $u,{u}^{\prime}$ be squares in $X$ with common vertices.

1.
A cubically thin homotopy $U:u{\equiv}_{T}^{\mathrm{\square}}{u}^{\prime}$ between $u$ and ${u}^{\prime}$ is a cube (http://planetmath.org/Polyhedron) $U\in {R}_{3}^{\mathrm{\square}}(X)$ such that

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$U$ is a homotopy^{} between $u$ and ${u}^{\prime},$
i.e. ${\partial}_{1}^{}(U)=u,{\partial}_{1}^{+}(U)={u}^{\prime},$

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$U$ is rel. vertices of ${I}^{2},$
i.e. ${\partial}_{2}^{}{\partial}_{2}^{}(U),{\partial}_{2}^{}{\partial}_{2}^{+}(U),{\partial}_{2}^{+}{\partial}_{2}^{}(U),{\partial}_{2}^{+}{\partial}_{2}^{+}(U)$ are constant,

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the faces ${\partial}_{i}^{\alpha}(U)$ are thin for $\alpha =\pm 1,i=1,2$.

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2.
The square $u$ is cubically $T$equivalent^{} to ${u}^{\prime},$ denoted $u{\equiv}_{T}^{\mathrm{\square}}{u}^{\prime}$ if there is a cubically thin homotopy between $u$ and ${u}^{\prime}.$
This definition enables one to construct the homotopy double groupoid^{} scheme ${\bm{\rho}}_{2}^{\mathrm{\square}}(X)$ , by defining a relation^{} of cubically thin homotopy on the set ${R}_{2}^{\mathrm{\square}}(X)$ of squares.
References
 1 K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2groupoid of a Hausdorff space, Applied Cat. Structures^{}, 8 (2000): 209234.
 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): 7193.
Title  cubically thin homotopy 
Canonical name  CubicallyThinHomotopy 
Date of creation  20130322 18:15:06 
Last modified on  20130322 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 