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cubically thin homotopy (Definition)

Cubically thin homotopy

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

  1. A cubically thin homotopy $U:u\equiv^{\square}_T u'$ between $u$ and $u'$ is a cube $U\in R^{\square}_3(X)$ such that
    • $U$ is a homotopy between $u$ and $u',$
      i.e. $\partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',$
    • $U$ 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,
    • the faces $\partial^{\alpha}_{i} (U) $ are thin for $\alpha = \pm 1, \ i = 1,2 $.
  2. The square $u$ is cubically $T$-equivalent to $u',$ denoted $u\equiv^{\square}_T u'$ if there is a cubically thin homotopy between $u$ and $u'.$

This definition enables one to construct $\boldsymbol{\rho}^{\square}_2 (X)$ , by defining a relation of cubically thin homotopy on the set $R^{\square}_2(X)$ of squares.

Bibliography

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.



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See Also: homotopy addition lemma and corollary


Cross-references: relation, homotopy, squares
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This is version 1 of cubically thin homotopy, born on 2009-03-03.
Object id is 559, canonical name is CubicallyThinHomotopy2.
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Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
Pending Errata and Addenda
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