1 Thin equivalence relation
Definition 1.1.
Let a,a′ : x ≃ y be paths in X. Then a is thinly equivalent to a′, denoted a ∼T a′, if there
is a thin relative homotopy between a and a′.
We note that ∼T is an equivalence relation, see [2]. We use ⟨a⟩ : x ≃ y to denote the ∼T
class of a path a : x ≃ y and call ⟨a⟩ the semitrack of a. The groupoid structure of
ρ1□(X) is induced by concatenation, +, of paths. Here one makes use of the fact that if
a : x ≃ x′, a′ : x′≃ x′′, a′′ : x′′≃ x′′′ are paths then there are canonical thin relative
homotopies
The source and target maps of ρ1□(X) are given by
if ⟨a⟩ : x ≃ y is a semitrack. Identities and inverses are given by
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.
