## You are here

Homethin square

## Primary tabs

# thin square

Let us consider first the concept of a tree that enters in the definition of a thin square. Thus, a simplified notion of thin square is that of “a continuous map from the unit square of the real plane into a Hausdorff space $X_{H}$ which factors through a tree” ([1]).

###### Definition 0.1.

A tree, is defined here as the underlying space $|K|$ of a
finite $1$-connected $1$-dimensional simplicial complex $K$ and
boundary $\partial{I}^{{2}}$ of $I^{{2}}=I\times I$ (that is, a *square* (interval) defined here as the Cartesian product of the unit interval $I:=[0,1]$ of real numbers).

###### Definition 0.2.

A *square map* $u:I^{{2}}\longrightarrow X$ in a topological space $X$ is *thin* if there
is a factorisation of $u$,

$u:I^{{2}}\stackrel{\Phi_{{u}}}{\longrightarrow}J_{{u}}\stackrel{p_{{u}}}{% \longrightarrow}X,$ |

where $J_{{u}}$ is a
*tree* and $\Phi_{{u}}$ is piecewise linear (PWL) on the
boundary $\partial{I}^{{2}}$ of $I^{{2}}$.

# References

- 1 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.
- 2
R. Brown and C.B. Spencer: Double groupoids and crossed modules,
*Cahiers Top. Géom.Diff.*, 17 (1976), 343–362. - 3 R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths Preprint, 1986.
- 4
K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff
*Applied Categorical Structures*, 8 (2000): 209-234. - 5
Al-Agl, F.A., Brown, R. and R. Steiner: 2002, Multiple categories: the equivalence of a globular and cubical approach,
*Adv. in Math*, 170: 711-118.

## Mathematics Subject Classification

18D05*no label found*55N33

*no label found*55N20

*no label found*55U40

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections