Thin equivalence relation

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

The source and target maps of $\boldsymbol{\rho}^\square_1 (X)$ are given by $$\partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1} \langle a\rangle =y,$$ if $\langle a\rangle :x\simeq y$ is a semitrack. Identities and inverses are given by $$\varepsilon (x)=\langle e_x\rangle \quad \mathrm{ resp.} -\langle a\rangle =\langle -a \rangle.$$

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.