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

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

     i.e. ${\partial }_1^{-}(U)=u,{\partial }_1^{+}(U)=u^{\prime },$

   • $U$ is rel. vertices of $I^2,$

     i.e. ${\partial }_2^{-}{\partial }_2^{-}(U),{\partial }_2^{-}{\partial }_2^{+}(U),{\partial }_2^{+}{\partial }_2^{-}(U),{\partial }_2^{+}{\partial }_2^{+}(U)$ areconstant,

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

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

This definition enables one to construct the homotopy double groupoid scheme ${𝝆}_2^{\mathrm{\square }}(X)$ , by defining arelation of cubically thin homotopy on the set $R_2^{\mathrm{\square }}(X)$ of squares.