PlanetPhysics/Thin Equivalence Relation

Thin equivalence relation

Let $a,a^{\prime }:x\simeq y$ be paths in $X$. Then $a$ is [[../ThinEquivalence/|thinly equivalent]] to $a^{\prime }$, denoted $a{\sim }_T{a}^{\prime }$, if there is a thin relative [[../ThinEquivalence/|homotopy]] between $a$ and $a^{\prime }$.

We note that ${\sim }_T$ is an [[../GroupoidHomomorphism2/|equivalence relation]], see . We use $⟨a⟩:x\simeq y$ to denote the ${\sim }_T$ class of a path $a:x\simeq y$ and call $⟨a⟩$ the {\it semitrack} of $a$. The [[../GroupoidHomomorphism2/|groupoid]] structure of ${\mathit{\rho }}_1^{◻}(X)$ is induced by concatenation, +, of paths. Here one makes use of the fact that if </math> a: x \simeq x', \ a' : x' \simeq x, \ a : x \simeq x' Failed to parse (unknown function "\begin{matrix}"): {\displaystyle are paths then there are canonical thin relative homotopies <center><math> \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 [[../SmallCategory/|target maps]] of ${\mathit{\rho }}_1^{◻}(X)$ are given by Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1} \langle a\rangle =y,} if $⟨a⟩:x\simeq y$ is a semitrack. [[../Cod/|Identities]] and inverses are given by $\epsilon (x)=⟨e_x⟩\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{.}-⟨a⟩=⟨-a⟩.$