cubically thin homotopy

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

  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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