# cubically thin homotopy

## 0.1 Cubically thin homotopy

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

1. 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

• $U$ is a homotopy between $u$ and $u^{\prime},$

i.e. $\partial^{-}_{1}(U)=u,\enskip\partial^{+}_{1}(U)=u^{\prime},$

• $U$ 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. 2.

The square $u$ is cubically $T$- 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