PlanetPhysics/Thin Equivalence Relation

Thin equivalence relation
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 . 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 {\it 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 $$ \begin{matrix}{r} (a+a') + a \simeq a+ (a' +a) : x \simeq x''' \ ({\it rescale}) \\ a+e_{x'} \simeq a:x \simeq x' ; \ e_{x} + a \simeq a: x \simeq x' \ ({\it dilation}) \\ a+(-a) \simeq e_{x} : x \simeq x \ ({\it cancellation}). \end{matrix} $$

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