0.1 Cubically thin homotopy
Let u,u′ be squares in X with common vertices.
- A cubically thin homotopy U : u ≡T □u′ between u and u′ is a cube U ∈ R
3□(X) such
that
- U is a homotopy between u and u′,
i.e. ∂1−(U) = u, ∂
1+(U) = u′,
- U is rel. vertices of I2,
i.e. ∂2−∂
2−(U), ∂
2−∂
2+(U), ∂
2+∂
2−(U), ∂
2+∂
2+(U) are constant,
- the faces ∂iα(U) are thin for α = ±1, i = 1, 2.
- The square u is cubically T-equivalent to u′, denoted u ≡T □u′ if there is a cubically thin
homotopy between u and u′.
This definition enables one to construct ρ2□(X) , by defining a relation of cubically thin homotopy
on the set R2□(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.
