thin equivalence relation

Thin equivalence relation

Definition 1.1  

Let $ a,a' : x \simeq y $ be paths in $ X $ . Then $ a$ is thinly equivalent to $ a' $ , denoted $ a \sim_{T} a' $ , if there is a thin relative homotopy between $ a $ and $ a' $ .

We note that $ \sim_{T} $ is an equivalence relation, see [2]. We use $ \langle a \rangle : x \simeq y $ to denote the $ \sim_{T} $ class of a path $ a: x \simeq y $ and call $ \langle a \rangle $ the semitrack of $ a $ . The groupoid structure of $ \boldsymbol{\rho}^\square_1 (X) $ is induced by concatenation, +, of paths. Here one makes use of the fact that if $ a: x \simeq
x', \ a' : x' \simeq x'', \ a'' : x'' \simeq x''' $ are paths then there are canonical thin relative homotopies

\begin{displaymath}
\begin{array}{r}
(a+a') + a'' \simeq a+ (a' +a'') : x \simeq...
...) \simeq e_{x} : x \simeq x \ ({\it cancellation}).
\end{array}\end{displaymath}

The source and target maps of $ \boldsymbol{\rho}^\square_1 (X)$ are given by

$\displaystyle \partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1}
\langle a\rangle =y,$

if $ \langle a\rangle :x\simeq y$ is a semitrack. Identities and inverses are given by

$\displaystyle \varepsilon (x)=\langle e_x\rangle \quad \mathrm{ resp.} -\langle a\rangle
=\langle -a \rangle.$

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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